A333990 - OEIS

%I #22 Sep 05 2020 03:42:10

%S 1,2,17,208,3281,62976,1419193,36643328,1064876737,34359869440,

%T 1217844546401,47005113741312,1961498610274321,87961440484327424,

%U 4217109422614386761,215187913985734475776,11641533109203575871233,665430291591787349803008,40065760383961956327231409

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

%H Seiichi Manyama, <a href="/A333990/b333990.txt">Table of n, a(n) for n = 0..381</a>

%F From _Vaclav Kotesovec_, Sep 05 2020: (Start)

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

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

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

%t 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 *)

%t Table[Hypergeometric2F1[1/2 - n, -n, 1/2, n], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 05 2020 *)

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

%Y Main diagonal of A333988.

%Y Cf. A333991.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 04 2020