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%I #26 Sep 08 2022 08:46:16
%S 0,0,6912,2073600,361267200,48771072000,5665247723520,595732271726592,
%T 58357447026278400,5420989989833932800,483204292920999936000,
%U 41671538221507034480640,3497929581885972295974912,287077554068924493987840000,23115688495680026711162880000
%N a(n) = 3*2^(2*n-1)*(n-1)*n^3*binomial(2*n,n)^2, a closed form for a triple binomial sum involving absolute values.
%C See Theorem 6 of Brent et al. article.
%C a(n) is divisible by 48^2.
%H Vincenzo Librandi, <a href="/A272913/b272913.txt">Table of n, a(n) for n = 0..500</a>
%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:1411.1477 [math.CO], 2016, page 11.
%F a(n) = Sum_{i=-n..n} (Sum_{j=-n..n} (Sum_{k=-n..n} binomial(2*n, n+i)*binomial(2*n, n+j)*binomial(2*n, n+k)*|(i^2-j^2)*(i^2-k^2)*(j^2-k^2)|)).
%F G.f.: 6912*x^2*(2F1(5/2, 5/2, 2, 64*x) + 100*x*2F1(7/2, 7/2, 3, 64*x)), where 2F1() is the Gauss hypergeometric function.
%F D-finite with recurrence (n-2)*(n-1)^2*a(n) -16*n*(2*n-1)^2*a(n-1)=0. - _R. J. Mathar_, Feb 08 2021
%t Table[3 2^(2 n - 1) (n - 1) n^3 Binomial[2 n, n]^2, {n, 0, 20}]
%o (PARI) vector(20, n, n--; 3*2^(2*n-1)*(n-1)*n^3*binomial(2*n,n)^2)
%o (Sage) [3*2^(2*n-1)*(n-1)*n^3*binomial(2*n, n)^2 for n in range(20)]
%o (Magma) [3*2^(2*n-1)*(n-1)*n^3*Binomial(2*n,n)^2: n in [0..20]];
%Y Cf. A254408, A268147, A268148, A268149, A268150, A268151, A268152, A269877.
%K nonn,easy
%O 0,3
%A _Bruno Berselli_, May 10 2016
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