OFFSET
0,2
REFERENCES
Frits Beukers, Some Congruences for Apery Numbers, Mathematisch Instituut, University of Leiden, 1983, pages 1-2.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..838
F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy
FORMULA
a(n) = 2F1([-n, 1 + 2*n], [1], -2), where 2F1 is the hypergeometric function. - Stefano Spezia, Dec 17 2020
From Vaclav Kotesovec, May 11 2021: (Start)
Recurrence: 3*n*(2*n - 1)*(26*n - 35)*a(n) = (2444*n^3 - 5734*n^2 + 3830*n - 729)*a(n-1) - (n-1)*(2*n - 3)*(26*n - 9)*a(n-2).
a(n) ~ sqrt(3/8 + 11/(8*sqrt(13))) * ((47 + 13*sqrt(13))/6)^n / sqrt(Pi*n). (End)
MATHEMATICA
Table[Sum[Binomial[n, k]*Binomial[2n+k, k]*2^k, {k, 0, n}], {n, 0, 20}] (* or *)
Table[Hypergeometric2F1[-n, 1+2 n, 1, -2], {n, 0, 20}] (* Stefano Spezia, Dec 17 2020 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ Michel Marcus, Feb 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Yifan Zhang, Dec 13 2020
EXTENSIONS
More terms from Stefano Spezia, Dec 17 2020
STATUS
approved