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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 19 11:36 EDT 2024. Contains 374394 sequences. (Running on oeis4.)