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%I #7 Dec 07 2024 05:42:15
%S 0,12,960,48384,2027520,76876800,2737373184,93351444480,3084788957184,
%T 99518467276800,3150448164864000,98221972499988480,
%U 3023952067480780800,92119659815689519104,2781153700681435054080,83317180181568395673600,2479232599432958230659072,73338095004533174933913600
%N a(n) = n^2 * 4^n * binomial(3*n, n).
%H Amiram Eldar, <a href="/A378779/b378779.txt">Table of n, a(n) for n = 0..500</a>
%H Necdet Batir, <a href="https://doi.org/10.1007/BF02829799">On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k</a>, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; <a href="https://arxiv.org/abs/math/0512310">arXiv preprint</a>, arXiv:math/0512310 [math.AC], 2005.
%F a(n) = A128782(n) * A005809(n).
%F a(n) = n^2 * A006588(n).
%F a(n) == 0 (mod 12).
%F Sum_{n>=1) (-1)^n/a(n) = 6 * arccot(2*sqrt(3)+sqrt(7))^2 - log(2)^2/2 (Batir, 2005, p. 379, eq. (3.8)).
%t a[n_] := n^2 * 4^n * Binomial[3*n, n]; Array[a, 25, 0]
%o (PARI) a(n) = n^2 * 4^n * binomial(3*n, n);
%Y Cf. A005809, A006588, A128782.
%K nonn,easy,new
%O 0,2
%A _Amiram Eldar_, Dec 07 2024