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a(n) = (n+1)^2*binomial(2*n+2,n-1)/2.
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%I #27 Jul 12 2024 10:17:12

%S 0,2,27,224,1500,8910,49049,256256,1288872,6298500,30093910,141210432,

%T 652860520,2981331990,13472983125,60343756800,268187306640,

%U 1183875281820,5194996380090,22676052360000,98513956031400,426171522716940,1836562780483002

%N a(n) = (n+1)^2*binomial(2*n+2,n-1)/2.

%F G.f.: 16*x*( 2-7*x +6*x*sqrt(1-4*x) )/ ( (1-4*x)^5/2 * (1+sqrt(1-4*x))^4 ). - _R. J. Mathar_, Nov 19 2011

%F a(n) = Sum_{k=0..n} k^2 * binomial(k+n, k). - _Stephen Bartell_, Jul 02 2024

%F a(n) ~ 2^(2*n+1)*n^(3/2)/sqrt(Pi). - _Stefano Spezia_, Jul 10 2024

%t Table[(n+1)^2 Binomial[2n+2,n-1]/2,{n,0,30}] (* _Harvey P. Dale_, Apr 15 2018 *)

%o (PARI) a(n) = (n+1)^2*binomial(2*n+2,n-1)/2 \\ _Michel Marcus_, Jun 08 2013

%K nonn

%O 0,2

%A _N. J. A. Sloane_, _Colin Mallows_