%I #50 Oct 13 2020 03:54:01

%S 6,60,420,2520,13860,72072,360360,1750320,8314020,38798760,178474296,

%T 811246800,3650610600,16287339600,72129646800,317370445920,

%U 1388495700900,6044040109800,26190840475800,113034153632400,486046860619320,2083057974082800,8900338616535600

%N a(n) = (2n+3)! /( n! * (n+1)! ).

%D E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99.

%H T. D. Noe, <a href="/A000911/b000911.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = 2 * A051133(n+1).

%F a(n) = A000984(n+1)*A000217(n). - _Zerinvary Lajos_, May 10 2007

%F a(n) = 6 * A002802(n). - _Zerinvary Lajos_, Jun 02 2007

%F n*a(n) - 2*(2*n+3)*a(n-1) = 0. - _R. J. Mathar_, Jun 07 2013

%F G.f.: 6*(1+10*x/( G(0)- 10*x)), where G(k)= 2*x*(2*k+5) + k + 1 - 2*x*(k+1)*(2*k+7)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 14 2013

%F Sum_{n>=0} (-1)^n/a(n) = 5*A086466-2 = 2*log(phi)*sqrt(5)-2 = 0.1520447... - _Jean-François Alcover_, Apr 22 2014

%F From _Ilya Gutkovskiy_, Jan 31 2017: (Start)

%F G.f.: 6/(1 - 4*x)^(5/2).

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

%F Sum_{n>=0} 1/a(n) = 2 - Pi/sqrt(3) = 2 - A093602. - _Amiram Eldar_, Oct 13 2020

%e 6 + 60*x + 420*x^2 + 2520*x^3 + 13860*x^4 + 72072*x^5 + 360360*x^6 + ...

%p seq(binomial(2*n,n)*binomial(n,(n-2)), n=2..21); # _Zerinvary Lajos_, May 10 2007

%t Table[(2 n + 3)!/(n!*(n + 1)!), {n, 0, 20}] (* _T. D. Noe_, Jun 20 2012 *)

%o (PARI) a(n) = 2^(n+4)*polcoeff(pollegendre(n+4),n) /* _Ralf Stephan_ */

%Y Cf. A000217, A000984, A001801, A002802, A051133, A086466, A093602.

%K nonn

%O 0,1

%A _N. J. A. Sloane_