A362729 a(n) = [x^n] ( E(x)/E(-x) )^n where E(x) = exp( Sum_{k >= 1} A108628(k-1)*x^k/k ).

1, 2, 8, 146, 1344, 18502, 214136, 2820834, 35377152, 465110894, 6038588808, 79936149174, 1056557893440, 14094461001558, 188319357861944, 2529143690991946, 34042038343081984, 459723572413090934, 6221522287903354568, 84397945280561045302, 1147007337762078241344

Offset

0,2

Comments

A108628(n) = B(n+1,n,n+1) in the notation of Straub, equation 24, where it is shown that the supercongruences A108628(n*p^k) == A108628(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.

Links

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

Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.

Formula

Conjecture:the supercongruence a(n*p^r) == a(n(p^(r-1)) (mod p^(3*r)) holds for all primes p >= 7 and positive integers n and r.

Maple

A108628 := proc(n) add(binomial(n, k)*binomial(n+1, k)*binomial(n+k+1, k) , k = 0..n) end:
E(n, x) := series(exp(n*add(2*(A108628(2*k)*x^(2*k+1))/(2*k+1), k = 0..10)), x, 21):
seq(coeftayl(E(n, x), x = 0, n), n = 0..20);

Crossrefs

Cf. A108628, A362722 - A362733.

Sequence in context: A124105 A079613 A091299 * A307326 A007314 A102099

Adjacent sequences: A362726 A362727 A362728 * A362730 A362731 A362732

Keyword

nonn,easy

Author

Peter Bala, May 03 2023

Status

approved