A334887 Denominator 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.

1, 1, 1, 1, 1, 5, 1, 35, 14, 210, 42, 462, 462, 6006, 858, 4290, 156, 145860, 29172, 2771340, 32604, 554268, 554268, 12748164, 12748164, 318704100, 63740820, 4574700, 191222460, 1459329300, 326203020, 34381798308, 343817983080, 68763596616, 343817983080, 12033629407800

Offset

0,6

Links

Table of n, a(n) for n=0..35.

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-> denom(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..37);  # Alois P. Heinz, May 14 2020

Mathematica

a[n_] := Denominator[2 * Sum[Binomial[n, k]^2 * Binomial[n + k, k]^2 * (HarmonicNumber[n + k] - HarmonicNumber[n - k]), {k, 0, n}]]; Array[a, 36, 0] (* Amiram Eldar, May 14 2020 *)

Prog

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

Crossrefs

Cf. A001008, A002805, A334886 (numerators).

Sequence in context: A336599 A066833 A379167 * A039813 A160632 A089515

Adjacent sequences: A334884 A334885 A334886 * A334888 A334889 A334890

Keyword

nonn,frac

Author

Michel Marcus, May 14 2020

Status

approved