OFFSET
0,2
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..300
FORMULA
a(n) = Sum_{k = 0..n} binomial(8*n, n-k)^2 * binomial(6*n+k-1, k).
a(n) = [x^n] (1 - x)^(2*n) * P(8*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial.
a(n) = (5/12)*(5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*(8*n-1)*(8*n-3)*(8*n-5)*(8*n-7)/((4*n-1)*(4*n-3)*(6*n-1)*(6*n-5)*n^2*(2*n-1)^2) * a(n-1) with a(0) = 1.
a(n) ~ c^n * sqrt(5)/(4*Pi*n), where c = (2^2)*(5^5)/(3^3).
a(p) == a(1) (mod p^3) for all primes p >= 5.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
a(n) = G(x)^(10*n), where the power series G(x) = 1 + 7*x + 412*x^2 + 55524*x^3 + 10088066*x^4 + 2146473322*x^5 + 503731865112*x^6 + ... appears to have integer coefficients
exp(Sum_{n >= 1} a(n)*x^n/n) = F(x)^10, where the power series F(x) = 1 + 7*x + 902*x^2 + 191779*x^3 + 50706776*x^4 + 15153397742*x^5 + 4898289306180*x^6 + ... appears to have integer coefficients.
MAPLE
seq( (8*n)!*(5*n)!*(3*n)! / ( (6*n)!*(4*n)!^2*n!^2 ), n = 0..13);
MATHEMATICA
A364305[n_]:=(8n)!(5n)!(3n)!/((6n)!(4n)!^2n!^2); Array[A364305, 15, 0] (* Paolo Xausa, Oct 06 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 21 2023
STATUS
approved