A069121 a(n) = n^4*binomial(2n,n).

0, 2, 96, 1620, 17920, 157500, 1197504, 8240232, 52715520, 318995820, 1847560000, 10328229912, 56073378816, 297051536600, 1541119305600, 7852824450000, 39392404439040, 194905125100620, 952671403252800

Offset: 0

Author

Benoit Cloitre, Apr 07 2002

Keywords

• easy
• nonn

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A002736.

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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

a(n) = a(n-1)*(4*n-2)*n^3/(n-1)^4, n>1. - Michael Somos, Apr 18 2003

Equals A002736*n^2. - Zerinvary Lajos, May 28 2006

From Ilya Gutkovskiy, Feb 07 2017: (Start)

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

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