A357558 a(n) = Sum_{k = 0..n} (-1)^(n+k)*k*binomial(n,k)*binomial(n+k,k)^2.

0, 4, 54, 648, 7500, 85440, 965202, 10849552, 121566744, 1359160020, 15172321890, 169175039616, 1884704860116, 20982512553912, 233474575117770, 2596777575029280, 28872014164369968, 320917108809011868, 3566175414049854306, 39620770883613043240, 440115513924937822020

Offset

0,2

Links

Table of n, a(n) for n=0..20.

A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014).

Formula

Conjecture: a(p-1) == 0 (mod (p - 1)*p^4) for all primes p >= 5 (checked up to p = 499).

Note: Let B(n) = A005258(n). It is known that B(n) = Sum_{k = 0..n} (-1)^(n+k)* binomial(n,k)*binomial(n+k,k)^2 and the supercongruences B(p-1) == 1 (mod p^3) hold for all primes p >= 5 (see, for example, Straub, Example 3.4).

Recurrence: a(0) = 0, a(1) = 4 and for n >= 2, (5*n - 2)*(n^2 - 1)*a(n) = (55*n^3 - 22*n^2 - 19*n + 10)*a(n-1) + n*(5*n + 3)*(n-1)*a(n-2).

a(n) ~ phi^(5*n + 7/2) / (2*Pi*5^(1/4)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 05 2022

Example

Example of a supercongruence:
p = 17: a(17 - 1) = 28872014164369968 = (2^4)*3*(17^4)*107*251*268153 == 0 (mod 16*7^4)

Maple

seq( add( (-1)^(n+k)*k*binomial(n, k)*binomial(n+k, k)^2, k = 0..n ), n = 0..20 );

Crossrefs

Cf. A005258, A357510, A357511, A357512, A357513, A357559, A357560, A357561.

Sequence in context: A269480 A199022 A221395 * A241126 A089205 A294041

Adjacent sequences: A357555 A357556 A357557 * A357559 A357560 A357561

Keyword

nonn,easy

Author

Peter Bala, Oct 03 2022

Status

approved