A254408 a(n) = 2*n^2*binomial(2*n,n)^2, a closed form for a double binomial sum involving absolute values.

0, 8, 288, 7200, 156800, 3175200, 61471872, 1154305152, 21201523200, 382952512800, 6826955907200, 120427502203008, 2105988385632768, 36562298361680000, 630861905459520000, 10827650254927680000, 184984389244186675200

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, and Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477 [math.CO], 2014.

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)).

From G. C. Greubel, Mar 31 2021: (Start)

a(n) = 8 * binomial(n+1, 2)^2 * C(n)^2, where C(n) = A000108(n) (Catalan numbers).

G.f.: 8*x*Hypergeometric2F1([3/2, 3/2], [1], 16*x) = (16/pi)*(x/(1-16*x)^2)*( 2*E(16*x) - (1-16*x)*K(16*x) ), where E(x) and K(x) are elliptic functions. (End)

D-finite with recurrence (n-1)^2*a(n) +(n^2-52*n+64)*a(n-1) -68*(2*n -3)^2*a(n-2)=0. - R. J. Mathar, Feb 27 2023

Maple

A254408:= n-> 2*( n*binomial(2*n, n) )^2; seq(A254408(n), n=0..30); # G. C. Greubel, Mar 31 2021

Mathematica

a[n_] := 2*n^2*Binomial[2*n, n]^2; Table[a[n], {n, 0, 20}]

Prog

(PARI) a(n) = 2*n^2*binomial(2*n, n)^2 \\ Charles R Greathouse IV, May 10 2016
(Magma) [(4*Binomial(n+1, 2)*Catalan(n))^2/2: n in [0..30]]; // G. C. Greubel, Mar 31 2021
(Sage) [(4*binomial(n+1, 2)*catalan_number(n))^2/2 for n in (0..30)] # G. C. Greubel, Mar 31 2021

Crossrefs

Cf. A000108, A000984, A002894.

Sequence in context: A187289 A187191 A054607 * A132592 A034977 A065141

Adjacent sequences: A254405 A254406 A254407 * A254409 A254410 A254411

Keyword

easy,nonn

Author

Jean-François Alcover, Jan 30 2015

Status

approved