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

Author

Andrew Howroyd, Feb 21 2021

Keywords

• nonn

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.

Crossrefs

Column k=3 of A146305.

Programs

Mathematica

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

PARI

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

Formula

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

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. - R. J. Mathar, Jul 31 2024

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. - R. J. Mathar, Jul 31 2024