A001762 - OEIS

%I M4741 N2029 #44 Mar 27 2024 21:00:20

%S 1,1,10,180,4620,152880,6168960,293025600,15990004800,984647664000,

%T 67493121696000,5094263446272000,419688934689024000,

%U 37465564582397952000,3601861863990534144000,370962724717928318976000,40744403224500159055872000

%N Number of labeled n-vertex dissections of a ball.

%C This is the number of labeled Apollonian networks (planar 3-trees). - _Allan Bickle_, Feb 20 2024

%D 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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001762/b001762.txt">Table of n, a(n) for n = 3..100</a>

%H L. W. Beineke and R. E. Pippert, <a href="https://doi.org/10.1007/BF02330563">The Number of Labeled Dissections of a k-Ball</a>, Math. Annalen, 191 (1971), 87-98.

%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.

%F a(n) = binomial(n,3)*(3*n-9)!/(2*n-4)!, n >= 4; a(3) = 1.

%F a(n) ~ 3^(3*n - 19/2) * n^(n-2) / (2^(2*n - 5/2) * exp(n)). - _Vaclav Kotesovec_, Mar 14 2024

%e There is one maximal planar graph with 4 vertices, and one way to label it, so a(4) = 1.

%t Join[{1}, Table[Binomial[n, 3]*(3*n - 9)!/(2*n - 4)!, {n, 4, 25}]] (* _T. D. Noe_, Aug 10 2012 *)

%o (Python)

%o from math import factorial

%o from sympy import binomial

%o def a(n):

%o     if n < 4:

%o         return 1

%o     else:

%o         return binomial(n, 3) * factorial(3*n-9) // factorial(2*n-4)

%o print([a(n) for n in range(3, 21)]) # _Paul Muljadi_, Mar 05 2024

%o (Julia)

%o using Combinatorics

%o a(n) = n < 4 ? 1 : binomial(BigInt(n),3)*factorial(BigInt(3*n-9))÷factorial(BigInt(2*n-4))

%o print([a(n) for n in 3:28]) # _Paul Muljadi_, Mar 27 2024

%Y Cf. A000407, A001763, A001764, A027610.

%K nonn

%O 3,3

%A _N. J. A. Sloane_

%E More terms from _Wolfdieter Lang_

%E Name clarified by _Andrey Zabolotskiy_, Mar 15 2024