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
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
KEYWORD
nonn,frac
AUTHOR
Michel Marcus, May 14 2020
STATUS
approved