A188685 Partial alternating sums of binomial(3n,n)^2/(2n+1)^2.

1, 0, 9, 135, 2890, 71639, 1967545, 58125959, 1813561210, 59034994415, 1987910416810, 68818255912790, 2437897047570874, 88061136002276310, 3234416650430634090, 120525771933269446806, 4548292982313797644875

Offset: 0

Author

Emanuele Munarini, Apr 08 2011

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..608

Crossrefs

Cf. A001764, A005809, A104859, A188678, A188681.

Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), A188683 (k=1), this sequence (k=2).

Programs

Magma

[ &+[(-1)^(n-k)*Binomial(3*k, k)^2/(2*k+1)^2: k in [0..n]]: n in [0..16]];  // Bruno Berselli, Apr 11 2011

Maple

A001764 := proc(n) binomial(3*n,n)/(2*n+1) ; end proc:
A188685 := proc(n) add( (-1)^(n-k)*A001764(k)^2,k=0..n) ; end proc: # R. J. Mathar, Apr 11 2011

Mathematica

Table[Sum[Binomial[3k,k]^2(-1)^(n-k)/(2k+1)^2,{k,0,n}],{n,0,20}]

Maxima

makelist(sum(binomial(3*k,k)^2*(-1)^(n-k)/(2*k+1)^2,k,0,n),n,0,20);

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*A001764(k)^2.

4*(2*n^2 + 9*n + 10)^2*a(n+2) - (713*n^4 + 4230*n^3 + 9317*n^2 + 9000*n + 3200)*a(n+1) - 9*(9*n^2 + 27*n + 20)^2*a(n) = 0.

a(n) ~ 3^(6*n+7)/(745*Pi*n^3*2^(4*n+4)). - Vaclav Kotesovec, Aug 06 2013