OFFSET
3,3
COMMENTS
This is the number of labeled Apollonian networks (planar 3-trees). - Allan Bickle, Feb 20 2024
REFERENCES
L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 3..100
L. W. Beineke and R. E. Pippert, The Number of Labeled Dissections of a k-Ball, Math. Annalen, 191 (1971), 87-98.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
FORMULA
a(n) = binomial(n,3)*(3*n-9)!/(2*n-4)!, n >= 4; a(3) = 1.
a(n) ~ 3^(3*n - 19/2) * n^(n-2) / (2^(2*n - 5/2) * exp(n)). - Vaclav Kotesovec, Mar 14 2024
EXAMPLE
There is one maximal planar graph with 4 vertices, and one way to label it, so a(4) = 1.
MATHEMATICA
Join[{1}, Table[Binomial[n, 3]*(3*n - 9)!/(2*n - 4)!, {n, 4, 25}]] (* T. D. Noe, Aug 10 2012 *)
PROG
(Python)
from math import factorial
from sympy import binomial
def a(n):
if n < 4:
return 1
else:
return binomial(n, 3) * factorial(3*n-9) // factorial(2*n-4)
print([a(n) for n in range(3, 21)]) # Paul Muljadi, Mar 05 2024
(Julia)
using Combinatorics
a(n) = n < 4 ? 1 : binomial(BigInt(n), 3)*factorial(BigInt(3*n-9))÷factorial(BigInt(2*n-4))
print([a(n) for n in 3:28]) # Paul Muljadi, Mar 27 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Wolfdieter Lang
Name clarified by Andrey Zabolotskiy, Mar 15 2024
STATUS
approved