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A378780 a(n) = n * 2^n * binomial(3*n, n).

%I A378780 #11 Mar 31 2025 02:55:23
%S A378780 0,6,120,2016,31680,480480,7128576,104186880,1506244608,21596889600,
%T A378780 307660953600,4359995228160,61522462310400,865005820084224,
%U A378780 12124867905454080,169509237023047680,2364380454476316672,32913250644698726400,457355892992216924160,6345297974846973542400
%N A378780 a(n) = n * 2^n * binomial(3*n, n).
%D A378780 Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.
%H A378780 Amiram Eldar, <a href="/A378780/b378780.txt">Table of n, a(n) for n = 0..500</a>
%H A378780 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. See p. 379, eq. (3.9).
%H A378780 Jonathan M. Borwein and Roland Girgensohn, <a href="https://doi.org/10.1007/s00010-005-2774-x">Evaluations of binomial series</a>, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 32, eq. (43).
%F A378780 a(n) = A036289(n) * A005809(n).
%F A378780 a(n) = n * A228484(n).
%F A378780 a(n) == 0 (mod 6).
%F A378780 Sum_{n>=1} 1/a(n) = Pi/10 - log(2)/5 (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005).
%t A378780 a[n_] := n * 2^n * Binomial[3*n, n]; Array[a, 25, 0]
%o A378780 (PARI) a(n) = n * 2^n * binomial(3*n, n);
%Y A378780 Cf. A005809, A036289, A228484.
%K A378780 nonn,easy
%O A378780 0,2
%A A378780 _Amiram Eldar_, Dec 07 2024