A119580 a(n) = (n^2+n^3)*binomial(2*n,n).

0, 4, 72, 720, 5600, 37800, 232848, 1345344, 7413120, 39382200, 203231600, 1024287264, 5062180032, 24607819600, 117942804000, 558423072000, 2615901857280, 12139419556440, 55866532906800, 255192804636000, 1157910842088000, 5222177897816880, 23422829664131040

Offset: 0

Author

Zerinvary Lajos, May 31 2006

Keywords

• easy
• nonn

Crossrefs

Cf. A000108, A000984, A001622, A011379.

Programs

Maple

[seq ((n^2+n^3)*(binomial(2*n,n)),n=0..29)];

Mathematica

Table[(n^2 + n^3) * Binomial[2 n, n], {n, 0, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)

Formula

From Amiram Eldar, Aug 28 2022: (Start)

a(n) = (n*(n+1))^2*A000108(n).

Sum_{n>=1} 1/a(n) = Pi/sqrt(3) - Pi^2/18 - 1.

Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(phi)^2 - 2*sqrt(5)*log(phi) + 1, where phi is the golden ratio (A001622). (End)

a(n) = A000984(n)*A011379(n). - Michel Marcus, Aug 28 2022