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%I #19 Feb 27 2023 08:17:10
%S 0,8,288,7200,156800,3175200,61471872,1154305152,21201523200,
%T 382952512800,6826955907200,120427502203008,2105988385632768,
%U 36562298361680000,630861905459520000,10827650254927680000,184984389244186675200
%N a(n) = 2*n^2*binomial(2*n,n)^2, a closed form for a double binomial sum involving absolute values.
%H G. C. Greubel, <a href="/A254408/b254408.txt">Table of n, a(n) for n = 0..500</a>
%H Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, and Helmut Prodinger, <a href="http://arxiv.org/abs/1411.1477">Some binomial sums involving absolute values</a>, arXiv:1411.1477 [math.CO], 2014.
%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)).
%F From _G. C. Greubel_, Mar 31 2021: (Start)
%F a(n) = 8 * binomial(n+1, 2)^2 * C(n)^2, where C(n) = A000108(n) (Catalan numbers).
%F 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)
%F 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
%p A254408:= n-> 2*( n*binomial(2*n, n) )^2; seq(A254408(n), n=0..30); # _G. C. Greubel_, Mar 31 2021
%t a[n_] := 2*n^2*Binomial[2*n, n]^2; Table[a[n], {n, 0, 20}]
%o (PARI) a(n) = 2*n^2*binomial(2*n,n)^2 \\ _Charles R Greathouse IV_, May 10 2016
%o (Magma) [(4*Binomial(n+1,2)*Catalan(n))^2/2: n in [0..30]]; // _G. C. Greubel_, Mar 31 2021
%o (Sage) [(4*binomial(n+1,2)*catalan_number(n))^2/2 for n in (0..30)] # _G. C. Greubel_, Mar 31 2021
%Y Cf. A000108, A000984, A002894.
%K easy,nonn
%O 0,2
%A _Jean-François Alcover_, Jan 30 2015
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