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

Author

Peter Bala, May 03 2023

Keywords

• nonn
• easy

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

• 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.

Crossrefs

Cf. A108628, A362722 - A362733.

Programs

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);

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.