A376463 a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)^2*A108625(n, k).

1, 13, 937, 110173, 16431001, 2815533013, 528281347609, 105661979187421, 22160058768609049, 4820836639111911013, 1079739020625352737937, 247635383880853678809541, 57923551410778898112945769, 13775523966484086307239139141, 3322958149149086403877851762937, 811467759428066412526078761086173

Offset: 0

Author

Peter Bala, Sep 24 2024

Keywords

• nonn
• easy

Comments

The sequence of Apéry numbers A005259 defined by A005259(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)^2 satisfies the pair of supercongruences

1) A005259(n*p^r) == A005259(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r

and

2) A005259(n*p^r - 1) == A005259(n*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r.

We conjecture that the present sequence satisfies the same pair of supercongruences. Some examples are given below.

Examples

			Examples of supercongruences:
a(11) - a(1) = 247635383880853678809541 - 13 = (2^3)*(3^2)*(11^3)*13*178697* 389023*2859343 == 0 (mod 11^3).
a(10) - a(0) = 1079739020625352737937 - 1 = (2^4)*3*(11^3)*13*1459*601831*1480561 == 0 (mod 11^3).
		

Crossrefs

Cf. A005259, A376458 - A376466.

Programs

Maple

A108625(n, k) := add(binomial(n, i)^2 * binomial(n+k-i, k-i), i = 0..k):
a(n) := add(binomial(n, k)^2*binomial(n+k, k)^2*A108625(n, k), k = 0..n):
seq(a(n), n = 0..25);