A188684 Partial sums of binomials binomial(3n,n)^2/(2n+1)^2.

1, 2, 11, 155, 3180, 77709, 2116893, 62210397, 1933897566, 62782453191, 2109727864416, 72915894194016, 2579631197677680, 93078664247524864, 3415556450680435264, 127175745034380516160, 4795994499281447607841

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. A005809, A001764, A188681.

Cf. Partial sums of binomial(3n,n)^2/(2n+1)^k: A188679 (k=0), A188682 (k=1), this sequence (k=2).

Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), A188683 (k=1), A188685 (k=2).

Programs

Magma

[&+[Binomial(3*k,k)^2/(2*k+1)^2: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 04 2016

Mathematica

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

Maxima

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

Formula

a(n) = sum( A001764(k)^2 , k=0..n).

4*(2*n^2+9*n+10)^2*a(n+2) - (745*n^4+4518*n^3+10285*n^2+10440*n+4000)*a(n+1) + 9*(9*n^2+27*n+20)^2*a(n) = 0.

a(n) = 4F3(1/3,1/3,2/3,2/3; 1,3/2,3/2; 729/16) - Gamma^2(3n+4) *5F4(1,n+4/3,n+4/3,n+5/3,n+5/3; n+2,n+2,n+5/2,n+5/2; 729/16)/ (Gamma(n+2)*Gamma(2n+3))^2, with pFq() generalized hypergeometric functions. - Charles R Greathouse IV, Apr 14 2011

a(n) ~ 3^(6*n+7)/(713*Pi*n^3*2^(4*n+4)). - Vaclav Kotesovec, Aug 06 2013