This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378777 #9 Dec 07 2024 05:42:48
%S A378777 0,3,60,756,7920,75075,668304,5697720,47070144,379632825,3004501500,
%T A378777 23417943120,180241588800,1372689900036,10360604899680,77595170756400,
%U A378777 577241321893632,4268838966063525,31404136939468020,229951212925133700,1676737802322198000,12180171012442098435
%N A378777 a(n) = n^2 * binomial(3*n, n).
%H A378777 Amiram Eldar, <a href="/A378777/b378777.txt">Table of n, a(n) for n = 0..500</a>
%H A378777 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 A378777 a(n) = n^2 * A005809(n).
%F A378777 a(n) = 3 * n * A090763(n-1) = 3 * n * A144485(n) / 2.
%F A378777 Sum_{n>=1} 1/a(n) = 6 * arctan(sqrt(3)/(2*phi-1))^2 - log((phi^3+1)/(phi+1)^3)^2/2, where phi = ((25+3*sqrt(69))/2)^(1/3) (Batir, 2005, p. 378, eq. (3.1)).
%t A378777 a[n_] := n^2 * Binomial[3*n, n]; Array[a, 25, 0]
%o A378777 (PARI) a(n) = n^2 * binomial(3*n, n);
%Y A378777 Cf. A005809, A090763, A144485.
%K A378777 nonn,easy
%O A378777 0,2
%A A378777 _Amiram Eldar_, Dec 07 2024