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

0,2

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

Table of n, a(n) for n=0..20.

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

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)

Prog

(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);

Crossrefs

Cf. A368961, A369513.

Cf. A151374.

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

Sequence in context: A354693 A112113 A368967 * A188266 A192625 A199561

Adjacent sequences: A369507 A369508 A369509 * A369511 A369512 A369513

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2024

Status

approved