A119580 - OEIS

%I #15 Aug 28 2022 08:29:13

%S 0,4,72,720,5600,37800,232848,1345344,7413120,39382200,203231600,

%T 1024287264,5062180032,24607819600,117942804000,558423072000,

%U 2615901857280,12139419556440,55866532906800,255192804636000,1157910842088000,5222177897816880,23422829664131040

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

%F From _Amiram Eldar_, Aug 28 2022: (Start)

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

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

%F 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)

%F a(n) = A000984(n)*A011379(n). - _Michel Marcus_, Aug 28 2022

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

%t Table[(n^2 + n^3) * Binomial[2 n, n], {n, 0, 30}] (* _Wesley Ivan Hurt_, Feb 26 2014 *)

%Y Cf. A000108, A000984, A001622, A011379.

%K easy,nonn

%O 0,2

%A _Zerinvary Lajos_, May 31 2006