537.sagews

sage: [abs(binomial(n+4,4)*(-1)^n) for n in xrange(-4, 19)] #nem kell A000332  	 	 Binomial coefficients binomial(n,4).

[0, 0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315]

sage: [binomial(n+4,4)*n/5 for n in xrange(1, 19)] #nem kell A000389  	 	 Binomial coefficients C(n,5).

[1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334]

sage: [binomial(n+4,4)*2^n for n in xrange(0, 19)] # nem kell  A003472  	 	 2^(n-4)*C(n,4).

[1, 10, 60, 280, 1120, 4032, 13440, 42240, 126720, 366080, 1025024, 2795520, 7454720, 19496960, 50135040, 127008768, 317521920, 784465920, 1917583360]

sage: [ceil(binomial(n+4,4)*2^n/10) for n in xrange(1, 19)] # nnnnnnnn

[1, 6, 28, 112, 404, 1344, 4224, 12672, 36608, 102503, 279552, 745472, 1949696, 5013504, 12700877, 31752192, 78446592, 191758336]

sage: [binomial(n+4,4)*3^n for n in xrange(0, 19)] #nem kellA036217    Expansion of 1/(1-3*x)^5; 5-fold convolution of A000244 (powers of 3).

[1, 15, 135, 945, 5670, 30618, 153090, 721710, 3247695, 14073345, 59108049, 241805655, 967222620, 3794488740, 14635885140, 55616363532, 208561363245, 772903875555, 2833980877035]

sage: [binomial(n+4,4)*4^n for n in xrange(0, 19)] #nem kell A040075  	 	 5-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^5.

[1, 20, 240, 2240, 17920, 129024, 860160, 5406720, 32440320, 187432960, 1049624576, 5725224960, 30534533120, 159719096320, 821412495360, 4161823309824, 20809116549120, 102821517066240, 502682972323840]

sage: [binomial(n+4,4)*5^n for n in xrange(0, 19)] #nem kell A03607 1   Expansion of 1/(1-5*x)^5. With a different offset, number of n-permutations (n=5) of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly four (4)u's. Example: a(1)=25 because we have uuuuv, uuuvu, uuvuu, uvuuu, vuuuu, uuuuw, uuuwu, uuwuu, uwuuu, wuuuu, uuuuz, uuuzu, uuzuu, uzuuu, zuuuu, uuuux, uuuxu, uuxuu, uxuuu, xuuuu uuuuy, uuuyu, uuyuu, uyuuu, yuuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008

[1, 25, 375, 4375, 43750, 393750, 3281250, 25781250, 193359375, 1396484375, 9775390625, 66650390625, 444335937500, 2905273437500, 18676757812500, 118286132812500, 739288330078125, 4566192626953125, 27904510498046875]

sage: [binomial(n+4,4)*6^n for n in xrange(0, 19)] #saaját nem kell A139626  	 	 Binomial(n+4,4)*6^n.    	 

With a different offset, number of n-permutations (n=5) of 7 objects t, u, v, w, z, x, y with repetition allowed, containing exactly four (4)u's. Example: a(1)=30 because we have
uuuut, uuutu, uutuu, utuuu, tuuuu,
uuuuv, uuuvu, uuvuu, uvuuu, vuuuu,
uuuuw, uuuwu, uuwuu, uwuuu, wuuuu,
uuuuz, uuuzu, uuzuu, uzuuu, zuuuu,
uuuux, uuuxu, uuxuu, uxuuu, xuuuu,
uuuuy, uuuyu, uuyuu, uyuuu, yuuuu.

[1, 30, 540, 7560, 90720, 979776, 9797760, 92378880, 831409920, 7205552640, 60526642176, 495217981440, 3961743851520, 31084451758080, 239794342133760, 1822437000216576, 13668277501624320, 101306056776744960, 742911083029463040]

sage: [binomial(n+4,4)*7^n for n in xrange(0, 19)] #nem kell saját A139641  	 	 Binomial(n+4,4)*7^n. 
With a different offset, number of n-permutations (n=5) of 8 objects s, t, u, v, w, z, x, y with repetition allowed, containing exactly four (4)u's. Example: a(1)=35 because we have
uuuus, uuusu, uusuu, usuuu, suuuu,  
....

[1, 35, 735, 12005, 168070, 2117682, 24706290, 271769190, 2853576495, 28852829005, 282757724249, 2699051004195, 25191142705820, 230595844768660, 2075362602917940, 18401548412539068, 161013548609716845, 1392293626213433895, 11911845468714934435]

sage: [n*k  for n in xrange (0, 22)for k in xrange(0, 5)] #

[0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 0, 3, 6, 9, 12, 0, 4, 8, 12, 16, 0, 5, 10, 15, 20, 0, 6, 12, 18, 24, 0, 7, 14, 21, 28, 0, 8, 16, 24, 32, 0, 9, 18, 27, 36, 0, 10, 20, 30, 40, 0, 11, 22, 33, 44, 0, 12, 24, 36, 48, 0, 13, 26, 39, 52, 0, 14, 28, 42, 56, 0, 15, 30, 45, 60, 0, 16, 32, 48, 64, 0, 17, 34, 51, 68, 0, 18, 36, 54, 72, 0, 19, 38, 57, 76, 0, 20, 40, 60, 80, 0, 21, 42, 63, 84]

sage: [sigma(1, 2*n)for n in xrange(0, 23)] #

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

sage: [sigma(2, 2*n)for n in xrange(0, 23)] #ok A052539  	   4^n + 1.

[2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417]

sage: [sigma(3,2*n)for n in xrange(0, 19)] #ok A062396    9^n + 1. context A083884

[2, 10, 82, 730, 6562, 59050, 531442, 4782970, 43046722, 387420490, 3486784402, 31381059610, 282429536482, 2541865828330, 22876792454962, 205891132094650, 1853020188851842, 16677181699666570, 150094635296999122]

sage: [sigma(3,2*n)/2for n in xrange(0, 19)] #A083884  	   a(n) = (3^(2n) + 1) / 2.    context A062396

[1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561]

sage: [sigma(4,2*n)for n in xrange(0, 19)] #nnnnnnnnn

[3, 21, 273, 4161, 65793, 1049601, 16781313, 268451841, 4295032833, 68719738881, 1099512676353, 17592190238721, 281474993487873, 4503599694479361, 72057594306363393, 1152921505680588801, 18446744078004518913, 295147905196532695041, 4722366482938364690433]

sage: [sigma(2, n)for n in xrange(0, 32)] #A000051  	 	 2^n + 1.

[2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649]

sage: [floor(sigma(2*n,n)/3) for n in xrange(1, 19)] #nnnnn

[1, 7, 84, 1456, 34386, 1012510, 35412392, 1437270101, 66252269583, 3416670271742, 194867871027996, 12176444816692219, 827151916103445674, 60690077505086538950, 4783115298130365934584, 402981422213394052000427, 36142954282075939563700998, 3438154723890713462651865097]

sage: [sigma(n,2)for n in xrange(1, 19)] #ok A001157    sigma_2(n): sum of squares of divisors of n.

[1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122, 210, 170, 250, 260, 341, 290, 455]

sage: [sigma(fibonacci(n),2)for n in xrange(1, 19)] #ok A063478  	 	 Sum_{d|F(n)} d^2, where F(n) are the Fibonacci numbers.

[1, 1, 5, 10, 26, 85, 170, 500, 1450, 3172, 7922, 31031, 54290, 143140, 483860, 1105000, 2550410, 8923300]

sage: [sigma(fibonacci(n),4)for n in xrange(1, 19)] #

[1, 1, 17, 82, 626, 4369, 28562, 196964, 1419874, 9165892, 62742242, 464378915, 2947295522, 20201388484, 147347450564, 961121685448, 6504586067282, 47555490592996]

sage: [sigma(2,fibonacci(n))for n in xrange(1, 9)] #nnnnnnnnn

[3, 3, 5, 9, 33, 257, 8193, 2097153]

sage: [sigma(fibonacci(n),3)for n in xrange(1, 19)] #nnnnnnnnnnn

[1, 1, 9, 28, 126, 585, 2198, 9632, 44226, 167832, 704970, 3543517, 12649338, 53609220, 257397588, 1000032768, 4073003174, 19720373400]

sage: [sigma(n,2)*sigma(2, n) for n in xrange(1, 19)] #nnnnnnnnnn

[3, 25, 90, 357, 858, 3250, 6450, 21845, 46683, 133250, 249978, 860370, 1392810, 4096250, 8519940, 22348117, 38011170, 119275975]

sage: [sigma(n,2)+sigma(2, n) for n in xrange(1, 19)] #nnnnnnnn

sage: [sigma(2,n)-sigma(n,2) for n in xrange(1, 19)] #nnnnnnnnnnnnnn

[2, 0, -1, -4, 7, 15, 79, 172, 422, 895, 1927, 3887, 8023, 16135, 32509, 65196, 130783, 261690]

sage: [2^n + 2^k for n in xrange (0, 1)for k in xrange(0, 10)] # sage: [sigma(2, n)for n in xrange(0, 32)] #A000051   2^n + 1.

[2, 3, 5, 9, 17, 33, 65, 129, 257, 513]

sage: [2^n + 2^k for n in xrange (0, 1)for k in xrange(0, 10)] #ok A000051  	 	 2^n + 1.  sage: [sigma(2, n)for n in xrange(0, 32)] #

[2, 3, 5, 9, 17, 33, 65, 129, 257, 513]

sage: [2^n *2^k for n in xrange (0, 1)for k in xrange(0, 10)] # sage: [sigma(2, n)for n in xrange(0, 32)] #

[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]

sage: [3^n *2^k for n in xrange (0, 1)for k in xrange(0, 10)] #

[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]

sage: [2^n *3^k for n in xrange (0, 1)for k in xrange(0, 10)] #

[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]

sage: [k^n *n^k for n in xrange (0, 1)for k in xrange(0, 10)] #

[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]

sage: [k^n -n^k for n in xrange (0, 1)for k in xrange(0, 10)] #

[0, 1, 1, 1, 1, 1, 1, 1, 1, 1]

sage: [k^(n+1) +n^(k+1) for n in xrange (0, 1)for k in xrange(0, 10)] #

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

sage: [floor(stirling_number1(n,1)/2) for n in xrange(0, 22)] #nnnnA001710  	 	 Order of alternating group A_n, or number of even permutations of n letters.
(Formerly M2933 N1179) 1, 1, 1, 3, 12, 60, 360

[0, 0, 0, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000]

sage: [floor(stirling_number1(n,2)/2) for n in xrange(0, 22)] #nnnnnnnnnnnnnnnnn

[0, 0, 0, 1, 5, 25, 137, 882, 6534, 54792, 513288, 5314320, 60271920, 743221440, 9901379520, 141732823680, 2169581500800, 35367141196800, 611702795289600, 11188494029260800, 215782573408819200, 4376474018380800000]

sage: [n^2*(n-1) for n in xrange(0, 40)] #ok A045991   n^3-n^2.

[0, 0, 4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, 2028, 2548, 3150, 3840, 4624, 5508, 6498, 7600, 8820, 10164, 11638, 13248, 15000, 16900, 18954, 21168, 23548, 26100, 28830, 31744, 34848, 38148, 41650, 45360, 49284, 53428, 57798]

sage: [n^2*(n-1)^2 for n in xrange(1, 35)] #ok A035287    Number of ways to place a non-attacking white and black rook on n X n chessboard.

[0, 4, 36, 144, 400, 900, 1764, 3136, 5184, 8100, 12100, 17424, 24336, 33124, 44100, 57600, 73984, 93636, 116964, 144400, 176400, 213444, 256036, 304704, 360000, 422500, 492804, 571536, 659344, 756900, 864900, 984064, 1115136, 1258884]

sage: [n^3*(n-2)^3 for n in xrange(1, 35)] #nnnnnn

[-1, 0, 27, 512, 3375, 13824, 42875, 110592, 250047, 512000, 970299, 1728000, 2924207, 4741632, 7414875, 11239424, 16581375, 23887872, 33698267, 46656000, 63521199, 85184000, 112678587, 147197952, 190109375, 242970624, 307546875, 385828352, 480048687, 592704000, 726572699, 884736000, 1070599167, 1287913472]

sage: [n^2*(n-2)^3 for n in xrange(1, 35)] #nnnnnnnnnnnn

[-1, 0, 9, 128, 675, 2304, 6125, 13824, 27783, 51200, 88209, 144000, 224939, 338688, 494325, 702464, 975375, 1327104, 1773593, 2332800, 3024819, 3872000, 4899069, 6133248, 7604375, 9345024, 11390625, 13779584, 16553403, 19756800, 23437829, 27648000, 32442399, 37879808]

sage: [n^2*(n-1)^3/4 for n in xrange(1, 35)] #A019584    n^2*(n-1)^3/4.

[0, 1, 18, 108, 400, 1125, 2646, 5488, 10368, 18225, 30250, 47916, 73008, 107653, 154350, 216000, 295936, 397953, 526338, 685900, 882000, 1120581, 1408198, 1752048, 2160000, 2640625, 3203226, 3857868, 4615408, 5487525, 6486750, 7626496, 8921088, 10385793]

sage: [n^2*(n-1)^2/4 for n in xrange(1, 35)] #o

[0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, 11025, 14400, 18496, 23409, 29241, 36100, 44100, 53361, 64009, 76176, 90000, 105625, 123201, 142884, 164836, 189225, 216225, 246016, 278784, 314721]

sage: [floor(stirling_number1(n,3)/2) for n in xrange(0, 22)] #nnnn

[0, 0, 0, 0, 3, 17, 112, 812, 6566, 59062, 586350, 6376788, 75458988, 965779776, 13298358528, 196078398912, 3082908807360, 51496122418560, 910801222312320, 17006124796911360, 334304865170576640, 6901879876820352000]

sage: [floor(stirling_number1(n,4)/2) for n in xrange(0, 22)] #nnnnnnnnnn

[0, 0, 0, 0, 0, 5, 42, 367, 3384, 33642, 361840, 4204750, 52629038, 707007444, 10156876548, 155494630200, 2528497851912, 43538874437952, 791656987863744, 15160627003859712, 305058037870245888, 6435465622575494400]

sage: [ceil(stirling_number1(n,1)/2) for n in xrange(1, 22)] #ok A001710  	 	 Order of alternating group A_n, or number of even permutations of n letters.

[1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000]

sage: [ceil(stirling_number1(n,2)/2) for n in xrange(1, 22)] #nnn

[0, 1, 2, 6, 25, 137, 882, 6534, 54792, 513288, 5314320, 60271920, 743221440, 9901379520, 141732823680, 2169581500800, 35367141196800, 611702795289600, 11188494029260800, 215782573408819200, 4376474018380800000]

sage: [ceil(stirling_number1(n,3)/2) for n in xrange(1, 22)] #nnn

[0, 0, 1, 3, 18, 113, 812, 6566, 59062, 586350, 6376788, 75458988, 965779776, 13298358528, 196078398912, 3082908807360, 51496122418560, 910801222312320, 17006124796911360, 334304865170576640, 6901879876820352000]

sage: [floor(fibonacci(n)/2) - 1 for n in xrange(3,29)]#nnnn

[0, 0, 1, 3, 5, 9, 16, 26, 43, 71, 115, 187, 304, 492, 797, 1291, 2089, 3381, 5472, 8854, 14327, 23183, 37511, 60695, 98208, 158904]

sage: [floor(sqrt((fibonacci(n)^3))) for n in xrange(1,51)]#nnnnnnnnn

[1, 1, 2, 5, 11, 22, 46, 96, 198, 407, 839, 1728, 3556, 7320, 15065, 31008, 63820, 131352, 270346, 556418, 1145204, 2357027, 4851165, 9984528, 20549866, 42295139, 87050631, 179165088, 368752393, 758955492, 1562060204, 3214987056, 6616993211, 13618903716, 28030033052, 57690601922, 118737125421, 244381311404, 502978534739, 1035215847540, 2130651264379, 4385244701564, 9025583592256, 18576194653800, 38232985633406, 78690023316765, 161957526125883, 333336287920860, 686063090136942, 1412035175000205]

sage: [floor(sqrt(fibonacci(n))) for n in xrange(0,51)]#ok A061287	  Integer part of square root of n-th Fibonacci number.

[0, 1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 31, 39, 50, 64, 82, 104, 133, 169, 215, 273, 348, 443, 563, 717, 912, 1160, 1475, 1877, 2388, 3037, 3863, 4915, 6252, 7952, 10116, 12867, 16368, 20820, 26484, 33688, 42852, 54508, 69336, 88197, 112188]

sage: [ceil(sqrt(fibonacci(n))) for n in xrange(0,41)]#nnnnnnnnnn

[0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 51, 65, 83, 105, 134, 170, 216, 274, 349, 444, 564, 718, 913, 1161, 1476, 1878, 2389, 3038, 3864, 4916, 6253, 7953, 10117]

sage: [ceil(binomial(n,3)/2) for n in xrange(0,41)]#nnnnn

[0, 0, 0, 1, 2, 5, 10, 18, 28, 42, 60, 83, 110, 143, 182, 228, 280, 340, 408, 485, 570, 665, 770, 886, 1012, 1150, 1300, 1463, 1638, 1827, 2030, 2248, 2480, 2728, 2992, 3273, 3570, 3885, 4218, 4570, 4940]

sage: [floor(binomial(n,4)/2) for n in xrange(0,41)]#nnn

[0, 0, 0, 0, 0, 2, 7, 17, 35, 63, 105, 165, 247, 357, 500, 682, 910, 1190, 1530, 1938, 2422, 2992, 3657, 4427, 5313, 6325, 7475, 8775, 10237, 11875, 13702, 15732, 17980, 20460, 23188, 26180, 29452, 33022, 36907, 41125, 45695]

sage: [floor(binomial(n,2)/3) for n in xrange(0,50)]#ok A062781   Number of arithmetic progressions of four terms and any mean which can be extracted from the set of the first n positive integers.

[0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392]

sage: [floor(binomial(n,3)/2) for n in xrange(0,41)]#ok A011894	  [ n(n-1)(n-2)/12 ].

[0, 0, 0, 0, 2, 5, 10, 17, 28, 42, 60, 82, 110, 143, 182, 227, 280, 340, 408, 484, 570, 665, 770, 885, 1012, 1150, 1300, 1462, 1638, 1827, 2030, 2247, 2480, 2728, 2992, 3272, 3570, 3885, 4218, 4569, 4940]

sage: [floor(binomial(n,2)/3) for n in xrange(-2,59)]#ok
A058937		 Maximal exponent of x in all terms of Somos polynomial of order n.

[1, 0, 0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551]

sage: [floor(binomial(n,2)/3) for n in xrange(0,60)]#ok
A130518		 Sum {0<=k<=n, floor(k/3)} (Partial sums of A002264).

[0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570]

sage: [floor(binomial(n,2)/3) for n in xrange(3,61)]#

[1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590]

sage: [floor(binomial(n,2)/3) for n in xrange(3,61)]#ok A130206  	   a(1) = 1, a(2) = 2; for n>2, a(n) = t(n)-a(n-1)-a(n-2), where t(n) = n(n+1)/2 = triangular number A000217.

[1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590]

sage: [floor(binomial(n,2)/3) for n in xrange(2,61)]#ok A001840		 Expansion of x/((1 - x)^2*(1 - x^3)).  seq(floor(binomial(n-1, 2)/3) , n=3..61); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 12 2009]

[0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590]

sage: [floor(binomial(n,2)/2) for n in xrange(0,58)]#ok A011848  [ C(n,2)/2 ].

[0, 0, 0, 1, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 203, 217, 232, 248, 264, 280, 297, 315, 333, 351, 370, 390, 410, 430, 451, 473, 495, 517, 540, 564, 588, 612, 637, 663, 689, 715, 742, 770, 798]

sage: [floor(binomial(n,3)/3) for n in xrange(0,60)]#duplikátum javítani A011849  	 	 [ C(n,3)/3 ].

[0, 0, 0, 0, 1, 3, 6, 11, 18, 28, 40, 55, 73, 95, 121, 151, 186, 226, 272, 323, 380, 443, 513, 590, 674, 766, 866, 975, 1092, 1218, 1353, 1498, 1653, 1818, 1994, 2181, 2380, 2590, 2812, 3046, 3293, 3553, 3826, 4113, 4414, 4730, 5060, 5405, 5765, 6141, 6533, 6941, 7366, 7808, 8268, 8745, 9240, 9753, 10285, 10836]

sage: [ceil(binomial(n,3)/3) for n in xrange(0,60)]#nnnnnnnnnnn

[0, 0, 0, 1, 2, 4, 7, 12, 19, 28, 40, 55, 74, 96, 122, 152, 187, 227, 272, 323, 380, 444, 514, 591, 675, 767, 867, 975, 1092, 1218, 1354, 1499, 1654, 1819, 1995, 2182, 2380, 2590, 2812, 3047, 3294, 3554, 3827, 4114, 4415, 4730, 5060, 5405, 5766, 6142, 6534, 6942, 7367, 7809, 8268, 8745, 9240, 9754, 10286, 10837]

sage: [floor(binomial(n,4)/4) for n in xrange(0,60)]#ok A011850[ C(n,4)/4 ].seq(floor(binomial(n, 4)/4), n=0.. 39); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 12 2009]

[0, 0, 0, 0, 0, 1, 3, 8, 17, 31, 52, 82, 123, 178, 250, 341, 455, 595, 765, 969, 1211, 1496, 1828, 2213, 2656, 3162, 3737, 4387, 5118, 5937, 6851, 7866, 8990, 10230, 11594, 13090, 14726, 16511, 18453, 20562, 22847, 25317, 27982, 30852, 33937, 37248, 40796, 44591, 48645, 52969, 57575, 62475, 67681, 73206, 79062, 85263, 91822, 98752, 106067, 113781]

sage: [floor(binomial(n,5)/5) for n in xrange(0,60)]#okA011851  [ C(n,5)/5 ]. seq(floor(binomial(n, 5)/5), n=0..37); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 12 2009]

[0, 0, 0, 0, 0, 0, 1, 4, 11, 25, 50, 92, 158, 257, 400, 600, 873, 1237, 1713, 2325, 3100, 4069, 5266, 6729, 8500, 10626, 13156, 16146, 19656, 23751, 28501, 33982, 40275, 47467, 55651, 64926, 75398, 87179, 100388, 115151, 131601, 149879, 170133, 192519, 217201, 244351, 274150, 306787, 342460, 381376, 423752, 469812, 519792, 573937, 632502, 695752, 763963, 837421, 916423, 1001277]

sage: [floor(binomial(n,6)/6) for n in xrange(0,37)]#ok  A011852  	 [ C(n,6)/6 ].

[0, 0, 0, 0, 0, 0, 0, 1, 4, 14, 35, 77, 154, 286, 500, 834, 1334, 2062, 3094, 4522, 6460, 9044, 12435, 16824, 22432, 29516, 38371, 49335, 62790, 79170, 98962, 122713, 151032, 184594, 224150, 270526, 324632]

sage: [floor(binomial(n,7)/7) for n in xrange(0,36)]#ok A011853    [ binomial(n,7)/7 ].

[0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 17, 47, 113, 245, 490, 919, 1634, 2778, 4546, 7198, 11074, 16611, 24363, 35022, 49443, 68671, 93971, 126861, 169148, 222968, 290828, 375653, 480836, 610292, 768516, 960645]

sage: [floor(binomial(n,8)/8) for n in xrange(0,36)]# ok A011854   [ binomial(n,8)/8 ].

[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 20, 61, 160, 375, 804, 1608, 3038, 5469, 9447, 15746, 25436, 39971, 61289, 91933, 135196, 195284, 277509, 388513, 536518, 731615, 986090, 1314787, 1735519, 2269525, 2941977]

sage: [floor(binomial(n,9)/9) for n in xrange(0,36)]#ok A011855  [ C(n,9)/9 ]. Zerinvary Lajos  [email protected]

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 24, 79, 222, 556, 1271, 2701, 5402, 10264, 18662, 32658, 55268, 90798, 145278, 226997, 347172, 520758, 767433, 1112778, 1589683, 2240008, 3116533, 4285233, 5827917, 7845273]

sage: [floor(binomial(n,10)/10) for n in xrange(0,36)]#ok A011856   [ C(n,10)/10 ].

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 28, 100, 300, 800, 1944, 4375, 9237, 18475, 35271, 64664, 114406, 196125, 326876, 531173, 843628, 1312311, 2003001, 3004501, 4435216, 6451224, 9256104, 13112814, 18357939]

sage: [floor(binomial(n,11)/11) for n in xrange(0,36)]#nnnnnnnnnnnnn

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 33, 124, 397, 1125, 2893, 6871, 15269, 32065, 64130, 122916, 226922, 405218, 702378, 1185263, 1952198, 3145208, 4966118, 7697483, 11729498, 17594247, 26008887, 37929627]

sage: [ceil(binomial(2*n,n)/4) for n in xrange(0,18)]#nnnnnnn

[1, 1, 2, 5, 18, 63, 231, 858, 3218, 12155, 46189, 176358, 676039, 2600150, 10029150, 38779380, 150270098, 583401555]

sage: [ceil(binomial(2*n,4)/2) for n in xrange(0,18)]#nnnn

[0, 0, 1, 8, 35, 105, 248, 501, 910, 1530, 2423, 3658, 5313, 7475, 10238, 13703, 17980, 23188]

sage: [ceil(binomial(n,2)/2) for n in xrange(0,58)]#ok A054925	 Ceiling(n*(n-1)/4).

[0, 0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53, 60, 68, 77, 86, 95, 105, 116, 127, 138, 150, 163, 176, 189, 203, 218, 233, 248, 264, 281, 298, 315, 333, 352, 371, 390, 410, 431, 452, 473, 495, 518, 541, 564, 588, 613, 638, 663, 689, 716, 743, 770, 798]

sage: [ceil(fibonacci(n)/2) for n in xrange(3,29)]#nnnn

[1, 2, 3, 4, 7, 11, 17, 28, 45, 72, 117, 189, 305, 494, 799, 1292, 2091, 3383, 5473, 8856, 14329, 23184, 37513, 60697, 98209, 158906]

sage: [floor(fibonacci(n)/2) for n in xrange(3,29)]#nnnn

[1, 1, 2, 4, 6, 10, 17, 27, 44, 72, 116, 188, 305, 493, 798, 1292, 2090, 3382, 5473, 8855, 14328, 23184, 37512, 60696, 98209, 158905]

sage: [ceil(6^6/2)  for n in xrange(1)]#

[23328]

sage: [ceil(n^n/3)  for n in xrange(0,21)]#

[1, 1, 2, 9, 86, 1042, 15552, 274515, 5592406, 129140163, 3333333334, 95103890204, 2972033482752, 100958368864085, 3704002275186006, 145964630126953125, 6148914691236517206, 275746753962112254726, 13115469358432179191808, 659473218553437863041327, 34952533333333333333333334]

sage: [ceil(n^n/2)  for n in xrange(0,21)]#ÚJ:  A168658 a(n) =Ceil(A000312(n)/2).

[1, 1, 2, 14, 128, 1563, 23328, 411772, 8388608, 193710245, 5000000000, 142655835306, 4458050224128, 151437553296127, 5556003412779008, 218946945190429688, 9223372036854775808, 413620130943168382089, 19673204037648268787712, 989209827830156794561990, 52428800000000000000000000]

sage: [floor(n^n/2)  for n in xrange(0,17)]#ok A057065  	 Floor[n^n/2].

[0, 0, 2, 13, 128, 1562, 23328, 411771, 8388608, 193710244, 5000000000, 142655835305, 4458050224128, 151437553296126, 5556003412779008, 218946945190429687, 9223372036854775808]

sage: [floor(n^3/2)  for n in xrange(0,41)]#A036487   [ (n^3)/2 ].

[0, 0, 4, 13, 32, 62, 108, 171, 256, 364, 500, 665, 864, 1098, 1372, 1687, 2048, 2456, 2916, 3429, 4000, 4630, 5324, 6083, 6912, 7812, 8788, 9841, 10976, 12194, 13500, 14895, 16384, 17968, 19652, 21437, 23328, 25326, 27436, 29659, 32000]

sage: [ceil(n^3/2)  for n in xrange(0,41)]#ok A036486   Ceiling (n^3)/2.

[0, 1, 4, 14, 32, 63, 108, 172, 256, 365, 500, 666, 864, 1099, 1372, 1688, 2048, 2457, 2916, 3430, 4000, 4631, 5324, 6084, 6912, 7813, 8788, 9842, 10976, 12195, 13500, 14896, 16384, 17969, 19652, 21438, 23328, 25327, 27436, 29660, 32000]

sage: [floor(3^n/2)  for n in xrange(0,27)]#ok beküldeni???A003462   (3^n - 1)/2.

[0, 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, 265720, 797161, 2391484, 7174453, 21523360, 64570081, 193710244, 581130733, 1743392200, 5230176601, 15690529804, 47071589413, 141214768240, 423644304721, 1270932914164]

sage: [ceil(5^n/2)  for n in xrange(0,21)]#ok A034478  	  (5^n+1)/2.

[1, 3, 13, 63, 313, 1563, 7813, 39063, 195313, 976563, 4882813, 24414063, 122070313, 610351563, 3051757813, 15258789063, 76293945313, 381469726563, 1907348632813, 9536743164063, 47683715820313]

sage: [ceil(3^n/2)  for n in xrange(0,26)]#ok A007051    (3^n + 1)/2.

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722]

sage: [floor(n/5)-floor(n/10)  for n in xrange(-5,98)]# ok A059995  	 	 Drop final digit of n.

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10]

Zerinvary Lajos  [e-mail]   [e-mail]

sage: [(factorial(n)/5) for n in xrange(1,20)]#nnnnnnnnnnnnnnn

[1/5, 2/5, 6/5, 24/5, 24, 144, 1008, 8064, 72576, 725760, 7983360, 95800320, 1245404160, 17435658240, 261534873600, 4184557977600, 71137485619200, 1280474741145600, 24329020081766400]

sage: [ceil(factorial(n)/5) for n in xrange(1,20)]#nnnnnnnnnnnnnn

[1, 1, 2, 5, 24, 144, 1008, 8064, 72576, 725760, 7983360, 95800320, 1245404160, 17435658240, 261534873600, 4184557977600, 71137485619200, 1280474741145600, 24329020081766400]

sage: [ceil(factorial(n)/7) for n in xrange(1,20)]#nnnnnnnnnn

[1, 1, 1, 4, 18, 103, 720, 5760, 51840, 518400, 5702400, 68428800, 889574400, 12454041600, 186810624000, 2988969984000, 50812489728000, 914624815104000, 17377871486976000]

sage: [floor(factorial(n)/5) for n in xrange(1,20)]#nnn

[0, 0, 1, 4, 24, 144, 1008, 8064, 72576, 725760, 7983360, 95800320, 1245404160, 17435658240, 261534873600, 4184557977600, 71137485619200, 1280474741145600, 24329020081766400]

sage: [floor(factorial(n)/7) for n in xrange(1,20)]#nnnn

[0, 0, 0, 3, 17, 102, 720, 5760, 51840, 518400, 5702400, 68428800, 889574400, 12454041600, 186810624000, 2988969984000, 50812489728000, 914624815104000, 17377871486976000]

sage: [floor(n/6) for n in xrange(0,90)]#ok A152467    Floor[n/2-n/3].

[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14]

sage: [floor(n/5) - 1 for n in xrange(5,88)]#ok A002266   Integers repeated 5 times.

[0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16]

sage: [floor(n/4) - 1 for n in xrange(4,84)]#ok A002265    Integers repeated 4 times.

[0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19]

sage: [floor(n/3) - 1 for n in xrange(3,79)]#ok A002264		 Integers repeated 3 times.

[0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25]

sage: [floor(n/2) + 1 for n in xrange(-1,75)]#ok A110654		 a(2*k) = k, a(2*k+1) = k+1.

[0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38]

sage: [floor(sqrt((binomial(n,2)^3))) for n in xrange(1,51)]#nnn

[0, 1, 5, 14, 31, 58, 96, 148, 216, 301, 407, 536, 688, 868, 1075, 1314, 1586, 1892, 2236, 2618, 3043, 3510, 4024, 4585, 5196, 5859, 6575, 7349, 8180, 9072, 10027, 11046, 12132, 13287, 14513, 15812, 17187, 18639, 20170, 21784, 23481, 25264, 27135, 29096, 31149, 33297, 35541, 37884, 40328, 42875]

sage: [ceil(sqrt(binomial(2*n,n))) for n in xrange(1,51)]#

[2, 3, 5, 9, 16, 31, 59, 114, 221, 430, 840, 1645, 3225, 6334, 12455, 24517, 48308, 95264, 188004, 371277, 733661, 1450552, 2869396, 5678698, 11243248, 22269229, 44124137, 87456793, 173399154, 343896179, 682223097, 1353744489, 2686899409, 5334139245, 10591802388, 21035982038, 41786727132, 83021808272, 164975800860, 327882919247, 651755000632, 1295727782450, 2576344908009, 5123329521092, 10189574146527, 20268089437391, 40319984379278, 80218869437723, 159617080097905, 317633978890113]

sage: [floor(sqrt(binomial(2*n,n))) for n in xrange(1,51)]#nnnnnnnnnnn

[1, 2, 4, 8, 15, 30, 58, 113, 220, 429, 839, 1644, 3224, 6333, 12454, 24516, 48307, 95263, 188003, 371276, 733660, 1450551, 2869395, 5678697, 11243247, 22269228, 44124136, 87456792, 173399153, 343896178, 682223096, 1353744488, 2686899408, 5334139244, 10591802387, 21035982037, 41786727131, 83021808271, 164975800859, 327882919246, 651755000631, 1295727782449, 2576344908008, 5123329521091, 10189574146526, 20268089437390, 40319984379277, 80218869437722, 159617080097904, 317633978890112]

sage: [ceil(sqrt(binomial(n,3))) for n in xrange(1,51)]#nnnnnnnnnnnnnnn

[0, 0, 1, 2, 4, 5, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 27, 29, 32, 34, 37, 40, 43, 45, 48, 51, 55, 58, 61, 64, 68, 71, 74, 78, 81, 85, 89, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140]

sage: [floor(sqrt(binomial(n,3))) for n in xrange(1,51)]#nnnnnn

[0, 0, 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 16, 19, 21, 23, 26, 28, 31, 33, 36, 39, 42, 44, 47, 50, 54, 57, 60, 63, 67, 70, 73, 77, 80, 84, 88, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 131, 135, 140]

complex_plot(zeta, (-30,5), (-8,8))

sage: [stirling_number2(3^n,2) for n in xrange(0,9)]#

[0, 3, 255, 67108863, 1208925819614629174706175, 7067388259113537318333190002971674063309935587502475832486424805170479103, 1412006979354108748474554421102313931675676955925788762341700965431346915180599249952936960497614998485448932749141998289061648432939195473813276544243473053215398045741358060286316036246351763861878679739417265182867455, 11260833093369905358008564931773237123962906992593153191740197879627764113449699015635737129588236155569065151454898685281992071828920434507861811564104229282355966195018235449999970199501725969385734062387706067008846278291494990910328688311146058890714818125840831682066623282620301901210654935936693089530528341925001296275329456002703269536816684104084956147550382553906528017946002652356208408650946852666095633807637309703882449659909643657820526881017004754429469363573583054334031860443041077960776477717795751511450374884097939090745853572483730643275070011585790025555679035760594463179762157452031299763564978560745342563505629468393329006177419263, 5711781108688968716417408782432195527460906738037201593101438093153597354379578310096407868428476580391352312660399399740246892891769535737654090779479204943069590253883127872634417647822558656752169409631870163441662732287133777748103847142934786157324158257164607965061375785732159431127989684886434721070408115256078600864640592078860693438657151913975527621763548654472053748097913636976467802338633883035773266723445348741627603918965231391342086074287840112598601743889952746226537815036179394836199267023287163696243086847008067061278879863113257504243399143450796231013595978495251148418413238452011531865046883676065172121009828101130444446013226947734114603494035960147504436689841223276684962447175035244445851005039270854112897617419547605544548854556081042560048823669061069916614055300041494696295669216888808272026842399823250206534885343543290743568751242826373857264720968098093001614588878533982060165115616020481929404419592144774112802515585138293859357521523203579767979813105527629648802605236097176077206154088189152989361021381135033586737475903043909907577039833429285293506360330711318908854898343708138018032105063646507099297725256848311816807497036717912421477041165107848843099280805202314783037395043284529821975617299908745161746026041112396954748594449962374014155336794381208841263674315332602708604679543331188902341888178256895294636295989478085153582558553794756044297661656084589812425744344519759704818709712847211993148119804650627751369465563500453062846064456423420402670615335012474320087476884386112185326522589624542242268721582301676430344089562194443835884745428980749486276292724187405568272668714746504623991136374069347475024909918743937674791378769202992426112687999102515947297042939614559839075050222051309528861101900249723744727553806748963793597432529837362772934923144086807220789088018791360837565259680974509438990741826175826594494083599357532239864803553897621713378472493127341259849907714676181708207678057086975]

sage: [stirling_number2(2^n,2) for n in xrange(0,9)]#A077585		 2^(2^n-1)-1.

[0, 1, 7, 127, 32767, 2147483647, 9223372036854775807, 170141183460469231731687303715884105727, 57896044618658097711785492504343953926634992332820282019728792003956564819967]

sage: [stirling_number1(n^2,1) for n in xrange(1, 22)] #

[1, 6, 40320, 1307674368000, 620448401733239439360000, 10333147966386144929666651337523200000000, 12413915592536072670862289047373375038521486354677760000000000, 1982608315404440064116146708361898137544773690227268628106279599612729753600000000000000, 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000, 933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000, 6689502913449127057588118054090372586752746333138029810295671352301633557244962989366874165271984981308157637893214090552534408589408121859898481114389650005964960521256960000000000000000000000000000, 38543707171800727705215657364933250819444321791546964384326881276202845420193798918144180166658987031965483631719296696351202501036957071818603525354815944336166154976340651887541505545310130488593669331551403376640000000000000000000000000000000000, 252607574497319838753801886917134114678628321660161899636045172255501630469951484784279524860515748677740723676917456344471493998101035608260664837215715659672830848592352788164518364067464570288846442542901808782229015393754650015551921726612213033664926565007360000000000000000000000000000000000000000, 2591899036156656511480987645585432937922090573952564787209169097471219681020477569926225190010791814894197085056341998303577816776333210349294122310324909594364025017659650820864712692975836305409649722536415268659156566243621875008684572005605493253978117505408263307569269403544381082593294633233365341569450141286400000000000000000000000000000000000000000000000, 55971593537537760404604573101364593176499404892579159768377152549395149245330647483833277915864388784447820896631966823753514906972682973565755801329326510071918742943565354051665634790924674413411085992886622125541764634019891159167420086648483483701916733257620551762703648325379440730697875571890344334786574299450407459134155346330065961388198972801957484316615567463363379200000000000000000000000000000000000000000000000000000, 3350850684932979117652665123754814942022584063591740702576779884286208799035732771005626138126763314259280802118502282445926550135522251856727692533193070412811083330325659322041700029792166250734253390513754466045711240338462701034020262992581378423147276636643647155396305352541105541439434840109915068285430675068591638581980604162940383356586739198268782104924614076605793562865241982176207428620969776803149467431386807972438247689158656000000000000000000000000000000000000000000000000000000000000000, 719676842666486216037680184173884806999005255738288188569731565112130658463427815835398303897420481427943461709105062477200567029251807542404607075317137505855720421327296621516773680762579338151716687101283909408016873091672124484509698673586140277672834972355244745093091892375183261604094435750963153352556353386468991311718730122664323846697527494860920004535541905687123518674808387136930648136929423429233135831620174991482991654252867370045587719220965382631816312264923378584493449925676161537934985695068160000000000000000000000000000000000000000000000000000000000000000000000, 706480697586142691712758304103567109576417070468951478933888078482146971006889065305293244784643284872314817727220226744120120416109490407725396294755403975112214116623172223826592905196815426174267515502858698320647960993570342580608984047032509201161639854951044437394946596042299788009514971950375708327784033231432348529145579855553040877907069191663366452046394277077878188231602347109790578465779826276654392916373853195895673138557258251616417221052811209627981870126429587134929665557183976222675820380101494720839043377444102395323273966041699239247035526675504711020305353480595308544000000000000000000000000000000000000000000000000000000000000000000000000000000, 3983166922118810678205990336564718434224120605664143701183608768141905250787782877719783678679061484962365081537093035923501297045164325783140744700987694957532389158400858116442933576455898753759925348456032608650278150327110180834816760596303097286523605280428429436106445252981359910377930818774855112452474332249873412259136132368731917415053983296042291396691439150433279600306468690306707682489567387757087354859605042563344537364880212053587269055695748062646674865811939840510090971364835065252717494840800357338638708817635670004576509737845399272402043442156827044195038802613229625783142492845706197046207004041551342095347194089863859659283335973442255940429503528960000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 160086307116559738155869925798757514626756457565007398646711114857005992922967078590696196618658161690735876437589642027120745407208793588072404971617007494843354135377095406066154855880767615610812537786121677226656934787295293329889991101773874178363226192550806087278026993983201987753863431668129069694725023374409414275815875828834913374670967078348380060934470394466978765779646756545675424549350157457563271478245865405680761395848801899028763255590217026083243137987131686080581096674871056010581499513879026589855942403498079792835159647491344925369568016515800543448680025803391561534522694855761493401748918989590240396787824784555716446448873404044136201133055019564546002121091038978073635688462008895936295056689750153498900363988015318027982295262581227520000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 56299448635663384211410188122316651841036248999122041309595746313692050999137594530505862585076461215223021983008336995777832707201314930846052067516257405228873705655160977518966047397236764107270500174639536046132430305499860256769856967603324233062680461494847143129164004203088468524557842011599069239234358828964614952717114638314523227959953631175758992769027159282689926311161722079132465458085006627266347113167723069061222036355742337137021139409054969586178877391803764752344391737585726326351570151035054536892557067261465331583222066323442251032729464305519070827691822662354457857702737748809675218225076724105367830616069025640211931604830222423561579520688824802412019881320370980787244157987803714384168607252725735326175062233683650422813765588005443658738188816755122239795323235301340793799955721464329162589567926440505632579000046624740581834752000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]

sage: [stirling_number1(n,1) for n in xrange(0, 22)] #ok A104150  	 	 Shifted factorial numbers: a(0)=0, a(n)=(n-1)!.

[0, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000]

sage: [stirling_number1(n,1) for n in xrange(1, 22)] #ok A000142  	 	 Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).

[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000]

sage: [stirling_number1(2*n,0) for n in xrange(1, 22)] #

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

sage: [stirling_number1(2*n,1) for n in xrange(1, 22)]#ok régen A009445		 (2n+1)!.

[1, 6, 120, 5040, 362880, 39916800, 6227020800, 1307674368000, 355687428096000, 121645100408832000, 51090942171709440000, 25852016738884976640000, 15511210043330985984000000, 10888869450418352160768000000, 8841761993739701954543616000000, 8222838654177922817725562880000000, 8683317618811886495518194401280000000, 10333147966386144929666651337523200000000, 13763753091226345046315979581580902400000000, 20397882081197443358640281739902897356800000000, 33452526613163807108170062053440751665152000000000]

sage: [stirling_number2(2*n+1,2) for n in xrange(1, 22)] #

[3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103]

sage: [stirling_number2(2*n,3)/6 for n in xrange(1, 22)] #nnnnnn

[0, 1, 15, 161, 1555, 14421, 131495, 1190281, 10739835, 96767741, 871346575, 7843866801, 70601791715, 635444087461, 5719108635255, 51472425109721, 463253615557195, 4169289698293581, 37523635917757535, 337712837792279041, 3039415998260356275]

sage: [stirling_number2(3*n,3) for n in xrange(1, 22)] #nnnnnnnn

[1, 90, 3025, 86526, 2375101, 64439010, 1742343625, 47063200806, 1270865805301, 34314651811530, 926505799458625, 25015738189761486, 675425583958589101, 18236495989562137650, 492385433499619572025, 13294407038741263288566, 358948992720026387542501, 9691622824832810693657370, 261673816441622674568827825, 7065193045292906500078982046, 190760212233861229795897561501]

sage: [stirling_number2(3*n,2) for n in xrange(1, 22)] #nnnnnn

[3, 31, 255, 2047, 16383, 131071, 1048575, 8388607, 67108863, 536870911, 4294967295, 34359738367, 274877906943, 2199023255551, 17592186044415, 140737488355327, 1125899906842623, 9007199254740991, 72057594037927935, 576460752303423487, 4611686018427387903]

sage: [stirling_number2(4*n,n) for n in xrange(0, 22)] #nnnn

[1, 1, 127, 86526, 171798901, 749206090500, 6090236036084530, 82892803728383735268, 1751346256720122175776157, 54294340536065700496358447625, 2364684125291482936353925428946680, 139762001313639974628848043262243505970, 10897986831117690497797320098390628446479030, 1095124825068104649239804362248176280579372065900, 139043013123786534660874830471520409000160718910717720, 21930813668442437111542041531912823008786998266055315277600, 4234713457698446178303023604485292647275809817872047264076226125, 988335457173212897880659074658497915819235657100670128536788951967175, 275678828267144840843200604174637129293662525054495080561160259129450181775, 90986687758406421985229785545067952190805327351821325668460466731882841914885000, 35216480564791974097669259465849601174356023206565782557760056606618405496605345727816, 15856875758539624950218589019418634240329835232788031556053371453402087420607455442547187456]

sage: [stirling_number2(n^2,n) for n in xrange(0, 12)] #nnnnnnn

[1, 1, 7, 3025, 171798901, 2436684974110751, 14204422416132896951197888, 50789872166903636182659702516635946082, 155440114706926165785630654089245708839702615196926765, 541500903058656141876322139677626107784896646583041951351456223689104719, 2754999986711164035029356262910003922476368243643133591265713197865860436127311130380917269755, 25545378351836929828979222353421914611063264552007039328326555030028639864663720016920351772612407113266112459931377276]

sage: [stirling_number1(n^2,n) for n in xrange(0, 12)] #nnnnnnn

[1, 1, 11, 118124, 5056995703824, 2677503356427960382362624, 43103055200236892507668550744976954163200, 44206966751754314698168885550132827351582613259130314424320000, 5404721354402004688383762084426637860253676685208119786101794968372441631679525683200000, 135545553634176928943936565148221939487754735614769129938030465900504564260226116951679073573163527073503694028800000000, 1125272380578944825172147216455781795628518533063359467753257922095126858974036298524724548786755944321693935796805099192994799803568595147738316800000000000, 4743241149101529188239038519016522502942041829190273397768220826120500562590923732686594709319031139975918370114381757186708051571875134410279091603872068741385500058589233442161950720000000000000000]

sage: [stirling_number1(n+2,n) for n in xrange(0, 12)] #
A000914 		Stirling numbers of first kind: s(n+2,n).

[0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717]

sage: [stirling_number1(n+5,n) for n in xrange(0, 12)] #	 PROGRAM   	
(Other) sage: [stirling_number1(n, n-5)*(-1)^(n+1) for n in xrange(6, 26)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]

[0, 120, 1764, 13132, 67284, 269325, 902055, 2637558, 6926634, 16669653, 37312275, 78558480]

sage: [stirling_number2(n+5,n) for n in xrange(0, 12)] #ok (n,n-5)

[0, 1, 63, 966, 7770, 42525, 179487, 627396, 1899612, 5135130, 12662650, 28936908]

sage: [stirling_number1(n+6,n) for n in xrange(0, 12)] #ok A112002 Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.

[0, 720, 13068, 118124, 723680, 3416930, 13339535, 44990231, 135036473, 368411615, 928095740, 2185031420]

sage: [stirling_number1(n+7,n) for n in xrange(0, 12)] #nnnnnnnnnn

[0, 5040, 109584, 1172700, 8409500, 45995730, 206070150, 790943153, 2681453775, 8207628000, 23057159840, 60202693980]

sage: [stirling_number2(2*n,2) for n in xrange(1, 22)] #A083420   a(n)=2*4^n-1.  	 MAPLE   	

[seq (stirling2(2*n, 2), n=1..23)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2006

[1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551]

sage: [stirling_number1(2*n,2) for n in xrange(1, 22)] #nnnnnnnnnnn

[1, 11, 274, 13068, 1026576, 120543840, 19802759040, 4339163001600, 1223405590579200, 431565146817638400, 186244810780170240000, 96538966652493066240000, 59190128811701203599360000, 42373564558110787183902720000, 35027999979859805266492784640000, 33115387462887740717065450291200000, 35504333673331019180373277094707200000, 42849305986961932992834955156389888000000, 57829595376179483326321825815950524416000000, 86763269335522422858175276287020406669312000000, 143943990158833766399457315931645601623572480000000]

age: [stirling_number1(2*n,3) for n in xrange(1, 22)] #

line 4 age: [stirling_number1(_sage_const_2 *n,_sage_const_3 ) for n in xrange(_sage_const_1 , _sage_const_22 )] # ^ SyntaxError: invalid syntax

sage: [stirling_number1(n,2) for n in xrange(1, 22)] #ok A000254  	 	 Stirling numbers of first kind s(n,2): a(n+1)=(n+1)*a(n)+n!. sage: [stirling_number1(i, 2) for i in xrange(1, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

[0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880, 19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200, 22376988058521600, 431565146817638400, 8752948036761600000]

sage: [stirling_number2(n,2) for n in xrange(1, 30)] #ok A000225  	 	 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.) #sage: [stirling_number2(i, 2) for i in xrange(1, 30)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

[0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455]

sage: [stirling_number1(n,3) for n in xrange(3, 22)] #ok A000399  	 	 Stirling numbers of first kind s(n,3).
sage: [stirling_number1(i+2, 3) for i in xrange(1, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

[1, 6, 35, 225, 1624, 13132, 118124, 1172700, 12753576, 150917976, 1931559552, 26596717056, 392156797824, 6165817614720, 102992244837120, 1821602444624640, 34012249593822720, 668609730341153280, 13803759753640704000]

sage: [stirling_number1(n,2)+stirling_number2(n,2) for n in xrange(3, 22)] #nnnnn

[6, 18, 65, 305, 1827, 13195, 109839, 1027087, 10629663, 120545887, 1486446975, 19802767231, 283465663743, 4339163034367, 70734282459135, 1223405590710271, 22376988058783743, 431565146818162687, 8752948036762648575]

sage: [stirling_number1(n,3)+stirling_number2(n,3) for n in xrange(3, 22)] #nnnnnnn

[2, 12, 60, 315, 1925, 14098, 121149, 1182030, 12782077, 151004502, 1931821177, 26597506026, 392159172925, 6165824756406, 102992266294945, 1821602509063650, 34012249787270821, 668609730921759726, 13803759755383047625]

sage: [stirling_number1(n,2)-stirling_number2(n,2) for n in xrange(3, 22)] #

[0, 4, 35, 243, 1701, 12941, 109329, 1026065, 10627617, 120541793, 1486438785, 19802750849, 283465630977, 4339162968833, 70734282328065, 1223405590448129, 22376988058259457, 431565146817114113, 8752948036760551425]

sage: [stirling_number1(n,3)-stirling_number2(n,3) for n in xrange(3, 22)] #nnnnnnn

[0, 0, 10, 135, 1323, 12166, 115099, 1163370, 12725075, 150831450, 1931297927, 26595928086, 392154422723, 6165810473034, 102992223379295, 1821602380185630, 34012249400374619, 668609729760546834, 13803759751898360375]

sage: [stirling_number1(n,2)*stirling_number2(n,2) for n in xrange(3, 22)] #nnnnnnnnnn

[9, 77, 750, 8494, 111132, 1659636, 27943920, 524580336, 10873098720, 246753240480, 6086983593600, 162204399296640, 4644017700698880, 142181354073427200, 4635571196664576000, 160352994162806323200, 5865970780625027788800, 226263996129579183820800, 9178122487647294720000000]

sage: [stirling_number1(n,3)*stirling_number2(n,3) for n in xrange(3, 22)] #nnnnnn

[1, 36, 875, 20250, 488824, 12685512, 357325100, 10941291000, 363489669576, 13058328791376, 505344267792000, 20984011855672320, 931412002668580224, 44034333337599217920, 2209989566072074464000, 117382258145191623206400, 6579605094663026514654720, 388199119294395373442042880, 24050892807787451154912000000]

sage: [stirling_number2(n,3) for n in xrange(0, 22)] #ok A000392  	 	 Stirling numbers of second kind S(n,3). #sage: [stirling_number2(i, 3) for i in xrange(0, 40)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

[0, 0, 0, 1, 6, 25, 90, 301, 966, 3025, 9330, 28501, 86526, 261625, 788970, 2375101, 7141686, 21457825, 64439010, 193448101, 580606446, 1742343625]

sage: [stirling_number1(n,n-1) for n in xrange(1, 55)] # A000217  	 	 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.

[0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431]

sage: [stirling_number2(n,n-1) for n in xrange(1, 22)] #

[0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210]

sage: [stirling_number1(n,n-2) for n in xrange(0, 38)] #ok 
A000914 		Stirling numbers of first kind: s(n+2,n).  (Other) SAGE:[stirling_number1(n+2, n)for n in xrange(0, 38)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 14 2009]

[0, 0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717, 3731, 5005, 6580, 8500, 10812, 13566, 16815, 20615, 25025, 30107, 35926, 42550, 50050, 58500, 67977, 78561, 90335, 103385, 117800, 133672, 151096, 170170, 190995, 213675, 238317, 265031, 293930, 325130, 358750, 394912, 433741, 475365, 519915, 567525, 618332, 672476, 730100, 791350, 856375, 925327, 998361]

sage: [stirling_number2(n,n-2) for n in xrange(2, 40)] # ok A001296  	 	 4-dimensional pyramidal numbers: (3n+1)*C(n+2,3)/4. Also Stirling2(n+2,n). Other) SAGE:[stirling_number2(n+2, n)for n in xrange(0, 38)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 14 2009]

[0, 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, 2431, 3367, 4550, 6020, 7820, 9996, 12597, 15675, 19285, 23485, 28336, 33902, 40250, 47450, 55575, 64701, 74907, 86275, 98890, 112840, 128216, 145112, 163625, 183855, 205905, 229881, 255892]

sage: [stirling_number1(n,n-3) for n in xrange(4, 34)] #ok A001303  	 	 Stirling numbers of first kind, s(n,n-3).

[6, 50, 225, 735, 1960, 4536, 9450, 18150, 32670, 55770, 91091, 143325, 218400, 323680, 468180, 662796, 920550, 1256850, 1689765, 2240315, 2932776, 3795000, 4858750, 6160050, 7739550, 9642906, 11921175, 14631225, 17836160, 21605760]

sage: [stirling_number2(n,n-3) for n in xrange(3, 34)] #ok 
A001297 		Stirling numbers of second kind S(n,n-3).

[0, 1, 15, 90, 350, 1050, 2646, 5880, 11880, 22275, 39325, 66066, 106470, 165620, 249900, 367200, 527136, 741285, 1023435, 1389850, 1859550, 2454606, 3200450, 4126200, 5265000, 6654375, 8336601, 10359090, 12774790, 15642600, 19027800]

sage: [stirling_number1(n,n-4) for n in xrange(5, 30)] # ok A000915  	 	 Stirling numbers of first kind s(n,n-4).

[24, 274, 1624, 6769, 22449, 63273, 157773, 357423, 749463, 1474473, 2749747, 4899622, 8394022, 13896582, 22323822, 34916946, 53327946, 79721796, 116896626, 168423871, 238810495, 333685495, 460012995, 626334345, 843041745]

sage: [stirling_number2(n,n-4) for n in xrange(5, 24)] #ok A001298  	 	 Stirling numbers of second kind.

[1, 31, 301, 1701, 6951, 22827, 63987, 159027, 359502, 752752, 1479478, 2757118, 4910178, 8408778, 13916778, 22350954, 34952799, 53374629, 79781779]

sage: [stirling_number1(n,n-5)*(-1)^(n+1) for n in xrange(6, 26)] #ok A053567  	 	 Stirling numbers of first kind.

[-120, 1764, -13132, 67284, -269325, 902055, -2637558, 6926634, -16669653, 37312275, -78558480, 156952432, -299650806, 549789282, -973941900, 1672280820, -2792167686, 4546047198, -7234669596, 11276842500]

sage: [stirling_number2(n,n-5) for n in xrange(6, 30)] #ok A112494  	 	 Sixth diagonal of the Stirling2 triangle A048993 and sixth column of triangle A008278.

[1, 63, 966, 7770, 42525, 179487, 627396, 1899612, 5135130, 12662650, 28936908, 62022324, 125854638, 243577530, 452329200, 809944464, 1404142047, 2364885369, 3880739170, 6220194750, 9759104355, 15015551265, 22693687380, 33738295500]

sage: [stirling_number1(n,n-6) for n in xrange(7, 27)] #ok A112002  	 	 Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.

[720, 13068, 118124, 723680, 3416930, 13339535, 44990231, 135036473, 368411615, 928095740, 2185031420, 4853222764, 10246937272, 20692933630, 40171771630, 75289668850, 136717357942, 241276443496, 414908513800, 696829576300]

sage: [stirling_number2(n,n-6) for n in xrange(6, 28)] #ok A144969  	 	 Stirling numbers of second kind S(n,n-6).

[0, 1, 127, 3025, 34105, 246730, 1323652, 5715424, 20912320, 67128490, 193754990, 512060978, 1256328866, 2892439160, 6302524580, 13087462580, 26046574004, 49916988803, 92484925445, 166218969675, 290622864675, 495564056130]

sage: [stirling_number1(n,n-7) for n in xrange(7, 27)] #nnnnnnn

[0, 5040, 109584, 1172700, 8409500, 45995730, 206070150, 790943153, 2681453775, 8207628000, 23057159840, 60202693980, 147560703732, 342252511900, 756111184500, 1599718388730, 3256091103430, 6400590336096, 12191224980000, 22563937825000]

sage: [stirling_number2(n,n-7) for n in xrange(7, 27)] #nnnnnnnnnnn

[0, 1, 255, 9330, 145750, 1379400, 9321312, 49329280, 216627840, 820784250, 2758334150, 8391004908, 23466951300, 61068660380, 149304004500, 345615943200, 762361127264, 1610949936915, 3275678594925, 6433839018750]

sage: [stirling_number2(2*n,n-1) for n in xrange(0, 14)] #nnnnnnnnn

[0, 0, 1, 31, 966, 34105, 1379400, 63436373, 3281882604, 189036065010, 12011282644725, 835143799377954, 63100165695775560, 5149507353856958820]

sage: [catalan_number(n) for n in xrange(0,27)]# - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152]

sage: [catalan_number(n)+stirling_number2(n,n-1) for n in range(27)]# nnnnnnn

[1, 1, 3, 8, 20, 52, 147, 450, 1458, 4898, 16841, 58841, 208078, 742978, 2674531, 9694950, 35357790, 129644926, 477638853, 1767263361, 6564120610, 24466267230, 91482563871, 343059613903, 1289904147600, 4861946401752, 18367353072477]

sage: [catalan_number(n)-stirling_number2(n,n-1) for n in range(27)]# n

[1, 1, 1, 2, 8, 32, 117, 408, 1402, 4826, 16751, 58731, 207946, 742822, 2674349, 9694740, 35357550, 129644654, 477638547, 1767263019, 6564120230, 24466266810, 91482563409, 343059613397, 1289904147048, 4861946401152, 18367353071827]

sage: [catalan_number(n)*stirling_number2(n,2) for n in xrange(2,20)]#nnnnnnn

[2, 15, 98, 630, 4092, 27027, 181610, 1239810, 8582756, 60138078, 425800564, 3042175500, 21906338040, 158830645635, 1158564772890, 8496271312650, 62604582047700, 463275674416170]

sage: [catalan_number(n)*stirling_number2(n,3) for n in xrange(3,20)]# nnnnnnnnnnn

[5, 84, 1050, 11880, 129129, 1381380, 14707550, 156706680, 1675459786, 17998446312, 194361212500, 2110052926800, 23026236054345, 252513376831620, 2781895215981750, 30778564965687000, 341873708072702190]

sage: [stirling_number2(n,n-6) for n in xrange(0, 28)] #

[0, 0, 0, 0, 0, 0, 0, 1, 127, 3025, 34105, 246730, 1323652, 5715424, 20912320, 67128490, 193754990, 512060978, 1256328866, 2892439160, 6302524580, 13087462580, 26046574004, 49916988803, 92484925445, 166218969675, 290622864675, 495564056130]

sage: [2*n/(1-n) for n in xrange(2, 28)] #

[-4, -3, -8/3, -5/2, -12/5, -7/3, -16/7, -9/4, -20/9, -11/5, -24/11, -13/6, -28/13, -15/7, -32/15, -17/8, -36/17, -19/9, -40/19, -21/10, -44/21, -23/11, -48/23, -25/12, -52/25, -27/13]

sage: [(n-1)*factorial(n)/2 for n in xrange(2, 21)] #ok A001286  	 	 Lah numbers: (n-1)*n!/2.

[1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000]

sage: [ceil((n+1)*factorial(n)/2) for n in xrange(0, 21)] #nnnnnn

[1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000]

sage: [floor((n+1)*factorial(n)/2) for n in xrange(0, 21)] #nnnnnnn

[0, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000]

sage: [ceil(catalan_number(n)*stirling_number2(n,2)/2) for n in xrange(2,20)]#nnnnnnnnnnnn

[1, 8, 49, 315, 2046, 13514, 90805, 619905, 4291378, 30069039, 212900282, 1521087750, 10953169020, 79415322818, 579282386445, 4248135656325, 31302291023850, 231637837208085]

sage: [floor(catalan_number(n)*stirling_number2(n,2)/2) for n in xrange(2,20)]#nnnnnnnnnnnnnnnnnnn

[1, 7, 49, 315, 2046, 13513, 90805, 619905, 4291378, 30069039, 212900282, 1521087750, 10953169020, 79415322817, 579282386445, 4248135656325, 31302291023850, 231637837208085]