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%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
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