A098772 a(n) = Sum_{k=0..n} binomial(2*n,2*k)^2.

1, 2, 38, 452, 6470, 92252, 1352540, 20056584, 300546630, 4537543340, 68923356788, 1052049129144, 16123803193628, 247959261273752, 3824345320438520, 59132290704871952, 916312070771835462, 14226520736453485260, 221256270142955957252, 3446310324328958045400, 53753604366737011495220

Offset: 0

Author

Vladeta Jovovic, Oct 03 2004

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A037964, A000984.

Programs

Mathematica

Table[Sum[Binomial[2n,2k]^2,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 21 2016 *)

Maxima

makelist((binomial(4*n,2*n)+(-1)^n*binomial(2*n,n))/2,n,0,12); /* Emanuele Munarini, Feb 01 2017 */

PARI

a(n) = sum(k=0, n, binomial(2*n, 2*k)^2); \\ Michel Marcus, Feb 01 2017

Formula

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

Recurrence: n*(n-1)*(2*n-1)*(5*n^2-15*n+11)*a(n)-4*(n-1)*(30*n^4-120*n^3+161*n^2-82*n+12)*a(n-1)-4*(4*n-7)*(2*n-3)*(4*n-5)*(5*n^2-5*n+1)*a(n-2) = 0.

a(n) ~ 2^(4*n-3/2)/sqrt(Pi*n). - Vaclav Kotesovec, Aug 02 2017