A268150 A double binomial sum involving absolute values.

0, 8, 2496, 177120, 7616000, 255780000, 7410154752, 194544814464, 4760448675840, 110493063252000, 2461297261280000, 53051182041906048, 1113060644163127296, 22833886572836393600, 459594580755139200000, 9100826722891800000000, 177680489488222659379200, 3426237501864596491802400

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..17.

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

Formula

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

Conjecture D-finite with recurrence -(4621*n-8921)*(n-1)^2*a(n) +4*(148256*n^3 -1055204*n^2 +2794799*n -2529792)*a(n-1) -64*(32443*n- 32400)*(2*n-3)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Feb 27 2023

Maple

A268150 := proc(n)
    add( add( binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2-l^2)^3, l=-n..n), k=-n..n) ;
end proc:
seq(A268150(n), n=0..10) ; # R. J. Mathar, Feb 27 2023

Prog

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

Crossrefs

Cf. A000984, A002894, A166337, A254408, A268148.

Sequence in context: A302952 A151580 A173175 * A325062 A247733 A343696

Adjacent sequences: A268147 A268148 A268149 * A268151 A268152 A268153

Keyword

easy,nonn

Author

Richard P. Brent, Jan 27 2016

Status

approved