A341853 Number of triangulations of a fixed pentagon with n internal nodes.
5, 21, 105, 595, 3675, 24150, 166257, 1186680, 8717940, 65572325, 502957455, 3922142574, 31021294850, 248377859100, 2010068042625, 16421073515280, 135277629836412, 1122788441510820, 9381874768828100, 78871575753345375, 666727830129370275
Offset: 0
Keywords
Examples
The a(0) = 5 triangulations correspond with the dissections of a pentagon by nonintersecting diagonals into 3 triangles. Although there is only one essentially different dissection, each rotation is counted separately here.
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..500
- K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
Crossrefs
Column k=2 of A146305.
Programs
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Mathematica
Array[210 Binomial[4 # + 5, #]/((3 # + 6)*(3 # + 7)) &, 21, 0] (* Michael De Vlieger, Feb 22 2021 *)
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PARI
a(n) = {210*binomial(4*n+5, n)/((3*n+6)*(3*n+7))}
Formula
a(n) = 210*binomial(4*n+5, n)/((3*n+6)*(3*n+7)).
D-finite with recurrence 3*n*(3*n+5)*(3*n+7)*(n+2)*a(n) -8*(4*n+5)*(2*n+1)*(4*n+3)*(n+1)*a(n-1)=0. - R. J. Mathar, Jul 31 2024
Comments