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%I #7 Dec 07 2024 05:39:15
%S 0,2,160,6048,168960,4004000,85542912,1701719040,32133218304,
%T 583116019200,10255365120000,175853140869120,2953078190899200,
%U 48728661198077952,792158036489666560,12713192776728576000,201760465448645689344,3170643145439310643200,49394436443159427809280
%N a(n) = n^3 * 2^n * binomial(3*n, n) / 3.
%H Amiram Eldar, <a href="/A378781/b378781.txt">Table of n, a(n) for n = 0..500</a>
%H D. V. Chudnovsky and G. V. Chudnovsky, <a href="https://doi.org/10.1073/pnas.95.6.2744">Classification of hypergeometric identities for Pi and other logarithms of algebraic numbers</a>, Proceedings of the National Academy of Sciences, Vol. 95, No. 6 (1998), pp. 2744-2749. See p. 2749.
%F a(n) = A128789(n) * A005809(n) / 3.
%F a(n) = n * A378778(n) / 3.
%F a(n) = n^2 * A378780(n) / 3.
%F Sum_{n>=1) 1/a(n) = 3*G*Pi - Pi^2*log(2)/8 + log(2)^3/2 - 99*zeta(3)/16, where G is Catalan's constant (Chudnovsky and Chudnovsky, 1998).
%t a[n_] := n^3 * 2^n * Binomial[3*n, n] / 3; Array[a, 25, 0]
%o (PARI) a(n) = n^3 * 2^n * binomial(3*n, n) / 3;
%Y Cf. A005809, A128789, A378778, A378780.
%Y Cf. A002117, A006752.
%K nonn,easy,new
%O 0,2
%A _Amiram Eldar_, Dec 07 2024