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a(n) = n^4*binomial(2n,n).
1

%I #22 May 24 2017 02:37:54

%S 0,2,96,1620,17920,157500,1197504,8240232,52715520,318995820,

%T 1847560000,10328229912,56073378816,297051536600,1541119305600,

%U 7852824450000,39392404439040,194905125100620,952671403252800

%N a(n) = n^4*binomial(2n,n).

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 386.

%H G. C. Greubel, <a href="/A069121/b069121.txt">Table of n, a(n) for n = 0..1000</a>

%F Sum_{n>=1} 1/a(n) = 17*Pi^4/3240. (Comtet, 1974)

%F a(n) = a(n-1)*(4*n-2)*n^3/(n-1)^4, n>1. - _Michael Somos_, Apr 18 2003

%F Equals A002736*n^2. - _Zerinvary Lajos_, May 28 2006

%F From _Ilya Gutkovskiy_, Feb 07 2017: (Start)

%F G.f.: 2*x*(1 + 30*x + 72*x^2 + 8*x^3)/(1 - 4*x)^(9/2).

%F a(n) ~ 4^n*n^(7/2)/sqrt(Pi). (End)

%p with(combinat):for n from 0 to 18 do printf(`%d, `,n^3*sum(binomial(2*n, n), k=1..n)) od: # _Zerinvary Lajos_, Mar 13 2007

%t Table[n^4*Binomial[2 n, n], {n, 0, 18}] (* or *)

%t CoefficientList[Series[2 x (1 + 30 x + 72 x^2 + 8 x^3)/(1 - 4 x)^(9/2), {x, 0, 18}], x] (* _Michael De Vlieger_, Feb 07 2017 *)

%o (PARI) a(n)=if(n<1,0,n^4*binomial(2*n,n))

%Y Cf. A002736.

%K easy,nonn

%O 0,2

%A _Benoit Cloitre_, Apr 07 2002