A007816 Number of abstract simplicial 2-complexes on {1,2,3,...,n+3} which triangulate the 2-sphere: C(n+3,2)*(4n+1)!/(3n+3)!.

1, 10, 195, 5712, 223440, 10929600, 641277000, 43859692800, 3424685806080, 300495408595200, 29262949937020800, 3131187613956864000, 365112996737448960000, 46075561988281233408000

Offset: 1

Author

Victor Reiner (reiner(AT)math.umn.edu), Paul Edelman (edelman(AT)math.umn.edu)

Keywords

• nonn
• easy
• nice

References

• Foulds, L. R. Enumeration of graph theoretic solutions for facilities layout. Proceedings of the sixteenth Southeastern international conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1985). Congr. Numer. 48 (1985), 87--99. MR0830702(87f:90045). See Table 2. - From N. J. A. Sloane, Apr 06 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

Programs

Magma

[Binomial(n+3, 2)*Factorial(4*n+1)/Factorial(3*n+3): n in [1..20]]; // Vincenzo Librandi, May 21 2012

Maple

A007816:=n->binomial(n+3,2)*(4*n+1)!/(3*n+3)!; seq(A007816(n), n=1..20);

Mathematica

Table[Binomial[n+3,2] (4n+1)!/(3n+3)!,{n,20}] (* Harvey P. Dale, May 16 2012 *)

Python

from math import factorial
def A007816(n): return ((n*(n + 5) + 6)*factorial((n<<2)+1)>>1)//factorial(3*(n+1)) # Chai Wah Wu, Nov 17 2022

Formula

a(n) ~ 2^(2+8*n)*3^(-7/2-3*n)*exp(-n)*n^n. - Stefano Spezia, Aug 03 2022