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

Author

Peter Bala, Sep 24 2024

Keywords

• nonn
• easy

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.

Examples

			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).
		

Crossrefs

Cf. A005258, 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)*A108625(n, n-k), k = 0..n):
seq(a(n), n = 0..25);