A376462 a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)*A108625(n, n-k).

1, 5, 109, 3317, 121501, 4954505, 216867925, 9981053045, 476860000285, 23451310381505, 1180189308268609, 60519806861966105, 3152285573768063461, 166371462775232899553, 8880340127444426907109, 478649327347386225075317, 26019989011889817463755805, 1425143757811438999747555313, 78578956793385528989609594089

Offset

0,2

Comments

The sequence of Apéry numbers A005258 defined by A005258(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k) 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..18.

Formula

a(n) ~ (2/3 + sqrt(31) * cos(arccos(1597/(434*sqrt(31)))/3)/6) * (19 + 28*sqrt(7/3) * cos(arccos(3*sqrt(3/7)/2)/3))^n / (Pi*n)^2. - Vaclav Kotesovec, Oct 16 2025

a(n) ~ (17 + 349/(4*(13*cos(Pi/7) - 8))) * 2^(7*n) * cos(Pi/7)^(7*n) / (26 * Pi^2 * n^2). - Vaclav Kotesovec, May 08 2026

Example

Examples of supercongruences:
a(11) - a(1) = 60519806861966105 - 5 = (2^2)*(3^2)*(5^2)*(11^3)*197*256454747 == 0 (mod 11^3).
a(10) - a(0) = 1180189308268609 - 1 = (2^6)*3*(11^3)*37*2789*44753 == 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)*A108625(n, n-k), k = 0..n):
seq(a(n), n = 0..25);

Mathematica

Table[Sum[Binomial[n, k]^2 * Binomial[n+k, k] * HypergeometricPFQ[{-n, k-n, n+1}, {1, 1}, 1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2025 *)

Crossrefs

Cf. A005258, A376458 - A376466.

Sequence in context: A012239 A012121 A371882 * A014180 A012122 A012091

Adjacent sequences: A376459 A376460 A376461 * A376463 A376464 A376465

Keyword

nonn,easy

Author

Peter Bala, Sep 24 2024

Status

approved