A334886 Numerator of 2*Sum_{k=0..n} binomial(n,k)^2*binomial(n+k,k)^2*(H(n+k)-H(n-k)) where H(n)=Sum_{k=1..n} 1/k.

0, 12, 210, 4438, 104825, 13276637, 70543291, 67890874657, 766399019471, 331100496521629, 1938622271127289, 632711558774104687, 18983793845583749117, 7475161875743183448469, 32609977844866666501811, 5014348056100175667015179, 5642363887250888549594351

Offset

0,2

Comments

Conjecture: a(p-1) == 0 (mod p^2) for all primes p >= 5 (checked up to p = 499). - Peter Bala, Oct 23 2022

Links

Table of n, a(n) for n=0..16.

Ira Gessel, Some congruences for Apéry numbers, Journal of Number Theory 14 (1982) 362-368. See b(n).

Eric Rowland, Reem Yassawi and Christian Krattenthaler, Lucas congruences for the Apéry numbers modulo p^2, arXiv:2005.04801 [math.NT], 2020. See A'(n).

Maple

H:= proc(n) option remember; `if`(n=0, 0, 1/n+H(n-1)) end:
a:= n-> numer(2*add(binomial(n, k)^2*binomial(n+k, k)^2*(H(n+k)-H(n-k)), k=0..n)):
seq(a(n), n=0..17);  # Alois P. Heinz, May 14 2020

Mathematica

a[n_] := Numerator[2 * Sum[Binomial[n, k]^2 * Binomial[n + k, k]^2 * (HarmonicNumber[n + k] - HarmonicNumber[n - k]), {k, 0, n}]]; Array[a, 17, 0] (* Amiram Eldar, May 14 2020 *)

Prog

(PARI) H(n) = sum(k=1, n, 1/k);
a(n) = numerator(2*sum(k=0, n, binomial(n, k)^2*binomial(n+k, k)^2*(H(n+k)-H(n-k))));

Crossrefs

Cf. A001008, A002805, A334887 (denominators).

Sequence in context: A129466 A259516 A342502 * A027399 A266910 A296681

Adjacent sequences: A334883 A334884 A334885 * A334887 A334888 A334889

Keyword

nonn,frac

Author

Michel Marcus, May 14 2020

Status

approved