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 A369510 #31 May 24 2024 00:43:55
%S A369510 1,4,28,240,2288,23296,248064,2728704,30764800,353633280,4128783360,
%T A369510 48827351040,583674642432,7041154416640,85610725769216,
%U A369510 1048040981594112,12907157115568128,159802897621319680,1987875305403187200,24833149969036738560,311409431144819589120
%N A369510 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).
%C A369510 a(n) also counts triangulations of a convex (2n+3)-gon whose points are colored red and blue alternatingly, and that do not have monochromatic triangles (i.e., every triangle has at least one red point and at least one blue point). - _Torsten Muetze_, May 08 2024
%D A369510 Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.
%H A369510 CombOS - Combinatorial Object Server, <a href="http://combos.org/kary">Generate k-ary trees and dissections</a>
%H A369510 Bruce E. Sagan, <a href="https://arxiv.org/abs/math/0407280">Proper partitions of a polygon and k-Catalan numbers</a>, arXiv:math/0407280 [math.CO], 2004.
%H A369510 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A369510 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(5*n+3,n-2*k).
%F A369510 From _Torsten Muetze_, May 08 2024: (Start)
%F A369510 a(n) = 2^n/(n+1) * binomial(3n+1,n).
%F A369510 a(n) = 2^n*A006013(n). (End)
%o A369510 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^2)^2)/x)
%o A369510 (PARI) a(n) = sum(k=0, n\2, binomial(2*n+k+1, k)*binomial(5*n+3, n-2*k))/(n+1);
%Y A369510 Cf. A368961, A369513.
%Y A369510 Cf. A151374.
%Y A369510 Cf. A153231 (colorful triangulations with an even number of points).
%K A369510 nonn
%O A369510 0,2
%A A369510 _Seiichi Manyama_, Jan 25 2024