A339710 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n + k, k)*2^k.

1, 7, 81, 1051, 14353, 201807, 2891409, 41976627, 615371169, 9089130967, 135048608401, 2016306678987, 30224723308081, 454603719479839, 6857319231939537, 103694587800440931, 1571449259865571137, 23860205774602899111, 362897293035114695121, 5527773456878667951483

Offset: 0

Author

Yifan Zhang, Dec 13 2020

Keywords

• nonn

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

Crossrefs

Cf. A000079 (Sum(binomial(n, k))), A000984 (Sum(binomial(n, k)^2)), A026375 (Sum(binomial(n, k)*binomial(2*k, k))), A001850 (Sum(binomial(n, k)*binomial(n+k, k))), A005809 (Sum(binomial(n, k)*binomial(2*n, k))).

Cf. A026000, A114496.

Programs

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 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ Michel Marcus, Feb 12 2021

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)

Extensions

More terms from Stefano Spezia, Dec 17 2020