A341854 Number of triangulations of a fixed hexagon with n internal nodes.

14, 84, 504, 3192, 21252, 147420, 1057224, 7791168, 58727214, 451137596, 3521923944, 27878644080, 223353322920, 1808402199660, 14778966163752, 121784464712064, 1011022286145816, 8449458780183120, 71042841567457760, 600615322603936920, 5103281457829103220

Offset

0,1

Comments

These may be called rooted [n,3] triangulations.

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], 2022-2023.

Formula

a(n) = 1008*binomial(4*n+7, n)/((3*n+8)*(3*n+9)).

From R. J. Mathar, Jul 31 2024: (Start)

D-finite with recurrence 3*(3*n+7)*(n+3)*(3*n+8)*a(n) + (-445*n^3-2164*n^2-3473*n-1838)*a(n-1) + 56*(4*n+1)*(2*n+1)*(4*n+3)*a(n-2) = 0.

D-finite with recurrence 3*n*(3*n+7)*(n+3)*(3*n+8)*a(n) - 8*(4*n+5)*(2*n+3)*(4*n+7)*(n+1)*a(n-1) = 0. (End)

a(n) ~ 7 * 2^(8*n+37/2) / (3^(3*n+15/2) * n^(5/2) * sqrt(Pi)). - Amiram Eldar, Oct 24 2025

Mathematica

Array[1008 Binomial[4 # + 7, #]/((3 # + 8) (3 # + 9)) &, 21, 0] (* Michael De Vlieger, Feb 22 2021 *)

Prog

(PARI) a(n) = {1008*binomial(4*n+7, n)/((3*n+8)*(3*n+9))}

Crossrefs

Column k=3 of A146305.

Sequence in context: A027818 A054149 A273182 * A025607 A272103 A244865

Adjacent sequences: A341851 A341852 A341853 * A341855 A341856 A341857

Keyword

nonn,easy

Author

Andrew Howroyd, Feb 21 2021

Status

approved