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

0,2

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.

Links

Table of n, a(n) for n=0..15.

Formula

a(n) ~ sqrt((68 + 117*cos(Pi/7)) / (7*(532*cos(Pi/7) - 479))) * (1 + 14/(10*cos(Pi/7) - 9))^n / (2^(5/2) * Pi^(5/2) * n^(5/2) * 5^n). - Vaclav Kotesovec, May 08 2026

Example

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).

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);

Mathematica

Table[Sum[Binomial[n, k]^2 * Binomial[n+k, k]^2 * HypergeometricPFQ[{-n, -k, n+1}, {1, 1}, 1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 08 2026 *)

Crossrefs

Cf. A005259, A376458 - A376466.

Sequence in context: A197068 A337575 A274544 * A267915 A358983 A096084

Adjacent sequences: A376460 A376461 A376462 * A376464 A376465 A376466

Keyword

nonn,easy

Author

Peter Bala, Sep 24 2024

Status

approved