A001762 Number of labeled n-vertex dissections of a ball. (Formerly M4741 N2029)

1, 1, 10, 180, 4620, 152880, 6168960, 293025600, 15990004800, 984647664000, 67493121696000, 5094263446272000, 419688934689024000, 37465564582397952000, 3601861863990534144000, 370962724717928318976000, 40744403224500159055872000

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

Cf. A000407, A001763, A001764, A027610.

Sequence in context: A113119 A067416 A113671 * A034908 A030048 A318796

Adjacent sequences: A001759 A001760 A001761 * A001763 A001764 A001765

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Wolfdieter Lang

Name clarified by Andrey Zabolotskiy, Mar 15 2024

Status

approved