A333990 a(n) = Sum_{k=0..n} n^k * binomial(2*n,2*k).

1, 2, 17, 208, 3281, 62976, 1419193, 36643328, 1064876737, 34359869440, 1217844546401, 47005113741312, 1961498610274321, 87961440484327424, 4217109422614386761, 215187913985734475776, 11641533109203575871233, 665430291591787349803008, 40065760383961956327231409

Offset: 0

Author

Seiichi Manyama, Sep 04 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..381

Crossrefs

Main diagonal of A333988.

Cf. A333991.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[n^k * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Sep 04 2020 *)
Table[Hypergeometric2F1[1/2 - n, -n, 1/2, n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 05 2020 *)

PARI

{a(n) = sum(k=0, n, n^k*binomial(2*n, 2*k))}

Formula

From Vaclav Kotesovec, Sep 05 2020: (Start)

a(n) = hypergeometric2F1(1/2 - n, -n, 1/2, n).

a(n) = (1 + sqrt(n))^(2*n)/2 + (1 - sqrt(n))^(2*n)/2.

a(n) ~ exp(2*sqrt(n) - 1) * n^n / 2 * (1 + 2/(3*sqrt(n))). (End)