A369513 -id:A369513 - cp's OEIS frontend

A369510 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).

1, 4, 28, 240, 2288, 23296, 248064, 2728704, 30764800, 353633280, 4128783360, 48827351040, 583674642432, 7041154416640, 85610725769216, 1048040981594112, 12907157115568128, 159802897621319680, 1987875305403187200, 24833149969036738560, 311409431144819589120

Offset: 0

Seiichi Manyama, Jan 25 2024

nonn

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

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

CombOS - Combinatorial Object Server, Generate k-ary trees and dissections
Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, arXiv:math/0407280 [math.CO], 2004.
Index entries for reversions of series

Cf. A368961, A369513.
Cf. A151374.
Cf. A153231 (colorful triangulations with an even number of points).

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

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(5*n+3,n-2*k).
From Torsten Muetze, May 08 2024: (Start)
a(n) = 2^n/(n+1) * binomial(3n+1,n).
a(n) = 2^n*A006013(n). (End)

cp's OEIS Frontend

A369510 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).

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