A376466 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)^2*A108625(n-1, k).

1, 3, 127, 9435, 866751, 89591753, 9988439203, 1173951006987, 143456999185855, 18063466831218981, 2329136571942011877, 306174745758226208537, 40896708938016175140963, 5536767359542664588001285, 758259747093486125157272779, 104880152366856305370319427435, 14632959744552362547801104612799

Offset

0,2

Comments

Cf. A005258(n) = Sum_{k = 0..n} (-1)^k*binomial(n, k)*binomial(n+k, k)^2.

The sequence of Apéry numbers A005258 satisfies the pair of supercongruences

1) A005258(n*p^r) == A005258(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r

and

2) A005258(n*p^r - 1) == A005258(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..16.

Formula

Examples of supercongruences:

a(11) - a(1) = 306174745758226208537 - 3 = 2*(11^3)*17*79367*85245689663 == 0 (mod 11^3).

a(10) - a(0) = 2329136571942011877 - 1 = (2^2)*(11^3)*17011*25717400209 == 0 (mod 11^3).

a(n) ~ 3^(3*n/2 - 1) / (Pi^2 * sin(Pi/18) * n^2 * 2^(9*n+3) * sin(Pi/9)^(9*n)). - Vaclav Kotesovec, May 08 2026

Maple

A108625(n, k) := add( binomial(n, i)^2 * binomial(n+k-i, k-i), i = 0..k):
a(n) := add((-1)^(n+k)*binomial(n, k)*binomial(n+k, k)^2*A108625(n-1, k), k = 0..n):
seq(a(n), n = 0..25);

Mathematica

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

Crossrefs

Cf. A005258, A376458 - A376465.

Sequence in context: A071151 A041867 A134711 * A163850 A258671 A243213

Adjacent sequences: A376463 A376464 A376465 * A376467 A376468 A376469

Keyword

nonn,easy

Author

Peter Bala, Sep 25 2024

Status

approved