3615.sagews

We examine several sequences that are congruent modulo p.  Only sequence A gives us supercongruences.

def A(n):
#This is sum of (2k choose k)^3 = truncated hypergeometric 1/2,1/2,1/2;1;1 evaluated at 64
    a=0
    for i in srange((n+1)/2):
        a = a + (binomial(2*i,i))^3
    return a

def B(n):
#This is sum of ((p-1)/2 choose i)^3*(-64)^i
    a=0
    for i in srange((n+1)/2):
        a = a + (binomial((n-1)//2,i))^3*(-64)^i
    return a

def C(n):
#This is sum of ((p-1)/2 choose i)^2*((p-1)/2+i choose i)*(64)^i
    a=0
    for i in srange((n+1)/2):
        a = a + (binomial((n-1)//2,i))^2*(binomial((n-1)//2+i,i))*(64)^i
    return a

def D(n):
#This is sum of ((p-1)/2 choose i)*((p-1)/2+i choose i)^2*(-64)^i
    a=0
    for i in srange((n+1)/2):
        a = a + (binomial((n-1)//2,i))*(binomial((n-1)//2+i,i))^2*(-64)^i
    return a

def E(n):
#This is sum of ((p-1)/2+i choose i)^3*(64)^i
    a=0
    for i in srange((n+1)/2):
        a = a + (binomial((n-1)//2+i,i))^3*(64)^i
    return a

#Here we compute some coefficients of the corresponding weight 3 modular form.

R.<q> = PowerSeriesRing(QQ)

eta=q
for n in srange(1,12):
    eta=eta*(1-q^(24*n))
eta+O(q^100)

q - q^25 - q^49 + O(q^100)

eta7=q^7
for n in srange(1,12):
    eta7=eta7*(1-q^(24*7*n))
eta7+O(q^700)

q^7 - q^175 - q^343 + O(q^700)

f=q^24
for n in srange(1,122):
    f=f*(1-q^(24*n))^3*(1-q^(24*7*n))^3+O(q^(24*121+1))
f

q^24 - 3*q^48 + 5*q^96 - 7*q^168 - 3*q^192 + 9*q^216 - 6*q^264 + 21*q^336 - 11*q^384 - 27*q^432 + 18*q^528 + 18*q^552 + 25*q^600 - 35*q^672 - 54*q^696 + 45*q^768 + 45*q^864 - 38*q^888 + 58*q^1032 - 30*q^1056 - 54*q^1104 + 49*q^1176 - 75*q^1200 - 6*q^1272 + 21*q^1344 + 162*q^1392 - 63*q^1512 - 91*q^1536 - 118*q^1608 + 114*q^1704 - 27*q^1728 + 114*q^1776 + 42*q^1848 - 94*q^1896 + 81*q^1944 - 174*q^2064 + 18*q^2112 + 90*q^2208 - 147*q^2352 - 54*q^2376 + 125*q^2400 + 18*q^2544 + 186*q^2568 + 106*q^2616 + 77*q^2688 - 222*q^2712 - 270*q^2784 - 85*q^2904 + O(q^2905)

#Here we look for congruences and supercongruences in our sequences.

p=11
print (A(p)%p^4).digits(p)
print (B(p)%p^4).digits(p)
print (C(p)%p^4).digits(p)
print (D(p)%p^4).digits(p)
print (E(p)%p^4).digits(p)

[5, 10, 1] [5, 1, 10, 7] [5, 7, 8] [5, 2, 10, 10] [5, 8, 3, 7]

p=5
print (A(p)%p^4).digits(p)
print (B(p)%p^4).digits(p)
print (C(p)%p^4).digits(p)
print (D(p)%p^4).digits(p)
print (E(p)%p^4).digits(p)

[0, 0, 4, 1] [0, 2, 3, 3] [0, 4, 3, 2] [0, 1, 2] [0, 3, 3, 1]

p=11
print ((A(3*p)+6*A(3))%p^6).digits(p)
print ((B(3*p)+6*B(3))%p^6).digits(p)
print ((C(3*p)+6*C(3))%p^6).digits(p)
print ((D(3*p)+6*D(3))%p^6).digits(p)
print ((E(3*p)+6*E(3))%p^6).digits(p)

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

p=11
print ((A(p^2)+6*A(p)+p^2)%p^8).digits(p)
print ((B(p^2)+6*B(p)+p^2)%p^8).digits(p)
print ((C(p^2)+6*C(p)+p^2)%p^8).digits(p)
print ((D(p^2)+6*D(p)+p^2)%p^8).digits(p)
print ((E(p^2)+6*E(p)+p^2)%p^8).digits(p)

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

p=11
print ((A(3*p^2)+6*A(3*p)+p^2*A(3))%p^8).digits(p)
print ((B(3*p^2)+6*B(3*p)+p^2*B(3))%p^8).digits(p)
print ((C(3*p^2)+6*C(3*p)+p^2*C(3))%p^8).digits(p)
print ((D(3*p^2)+6*D(3*p)+p^2*D(3))%p^8).digits(p)
print ((E(3*p^2)+6*E(3*p)+p^2*E(3))%p^8).digits(p)

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