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A369510
Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).
3
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.
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. A151374.
Cf. A153231 (colorful triangulations with an even number of points).
Sequence in context: A354693 A112113 A368967 * A188266 A192625 A199561
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 25 2024
STATUS
approved