A268150 - OEIS

%I #22 Feb 27 2023 08:03:41

%S 0,8,2496,177120,7616000,255780000,7410154752,194544814464,

%T 4760448675840,110493063252000,2461297261280000,53051182041906048,

%U 1113060644163127296,22833886572836393600,459594580755139200000,9100826722891800000000,177680489488222659379200,3426237501864596491802400

%N A double binomial sum involving absolute values.

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

%H Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, <a href="http://arxiv.org/abs/1411.1477">Some binomial sums involving absolute values</a>, arXiv:1411.1477v2 [math.CO], 2016.

%F 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).

%F 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

%p A268150 := proc(n)

%p     add( add( binomial(2*n,n+k)*binomial(2*n,n+l)*abs(k^2-l^2)^3,l=-n..n),k=-n..n) ;

%p end proc:

%p seq(A268150(n),n=0..10) ; # _R. J. Mathar_, Feb 27 2023

%o (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));

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

%K easy,nonn

%O 0,2

%A _Richard P. Brent_, Jan 27 2016