A268149 A double binomial sum involving absolute values.

0, 24, 1120, 33264, 823680, 18475600, 389398464, 7862853600, 153876579840, 2940343837200, 55138611528000, 1018383898440480, 18574619721465600, 335240928272918304, 5996573430996184960, 106438123408375281600, 1876607120325212706816, 32891715945378106711440

Offset: 0

Author

Richard P. Brent, Jan 27 2016

Keywords

• easy
• nonn

Comments

A fast algorithm follows from Theorem 1 of Brent et al. article.

Links

• Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477v2 [math.CO], 2016.

Crossrefs

Cf. A000984, A002894, A166337, A268147.

Programs

PARI

a(n) = sum(k=-n,n, sum(l=-n,n, binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k-l)^3));

Formula

a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k-l)^3).

Conjecture D-finite with recurrence (2*n-1)*(n-1)*a(n) +2*(-22*n^2+27*n-36)*a(n-1) +12*(4*n-5)*(4*n-7)*a(n-2)=0. - R. J. Mathar, Feb 27 2023