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

Author

Seiichi Manyama, Jan 25 2024

Keywords

• nonn

Comments

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

References

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

Links

• 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

Crossrefs

Cf. A368961, A369513.

Cf. A151374.

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

Programs

PARI

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

PARI

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

Formula

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)