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
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
KEYWORD
easy,nonn
AUTHOR
Jean-François Alcover, Jan 30 2015
STATUS
approved