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Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).
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%I #31 May 24 2024 00:43:55

%S 1,4,28,240,2288,23296,248064,2728704,30764800,353633280,4128783360,

%T 48827351040,583674642432,7041154416640,85610725769216,

%U 1048040981594112,12907157115568128,159802897621319680,1987875305403187200,24833149969036738560,311409431144819589120

%N Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).

%C 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 Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.

%H CombOS - Combinatorial Object Server, <a href="http://combos.org/kary">Generate k-ary trees and dissections</a>

%H 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 <a href="/index/Res#revert">Index entries for reversions of series</a>

%F 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 From _Torsten Muetze_, May 08 2024: (Start)

%F a(n) = 2^n/(n+1) * binomial(3n+1,n).

%F a(n) = 2^n*A006013(n). (End)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^2)^2)/x)

%o (PARI) a(n) = sum(k=0, n\2, binomial(2*n+k+1, k)*binomial(5*n+3, n-2*k))/(n+1);

%Y Cf. A368961, A369513.

%Y Cf. A151374.

%Y Cf. A153231 (colorful triangulations with an even number of points).

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 25 2024