A000287 Number of rooted polyhedral graphs with n edges.

1, 0, 4, 6, 24, 66, 214, 676, 2209, 7296, 24460, 82926, 284068, 981882, 3421318, 12007554, 42416488, 150718770, 538421590, 1932856590, 6969847486, 25237057110, 91729488354, 334589415276, 1224445617889, 4494622119424

Offset: 6

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice

Comments

a(n) appears to be odd if and only if n = 2^k - 2 for some integer k >= 3. - Lewis Chen, May 05 2019

Examples

			G.f. = x^6 + 4*x^8 + 6*x^9 + 24*x^10 + 66*x^11 + 214*x^12 + 676*x^13 + ...
		

References

• Handbook of Combinatorics, North-Holland '95, p. 892. (Gives different last term)
• 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).
• W. T. Tutte, Three-connected planar maps. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 43--52. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0335323 (49 #105). - From N. J. A. Sloane, Jun 05 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 6..1000
• A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs, Math. Comp. 37 (1981), no. 156, 523-532.
• Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
• W. T. Tutte, A new branch of enumerative graph theory, Bull. Amer. Math. Soc., 68 (1962), 500-504.
• W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
• Liu Yanpei, On the number of rooted c-nets, J. Combin. Theory, B 36 (1984), 118-123.

Crossrefs

Cf. A000256.

Programs

Mathematica

a[6] = 1; a[n_] := a[n] = ((9*(5 - 3*n)*n - 16)*a[n-1]*((n-1)!)^2 + 2*((-1)^n*(9*n*(3*n - 17) + 160)*((n-1)!)^2 + ((2*n - 2)!)))/(2*(9*n*(3*n - 11) + 88)*((n-1)!)^2); Table[ a[n], {n, 6, 31}] (* Jean-François Alcover, Oct 04 2011, after formula *)

PARI

seq(N) = {
  my(x='x+O('x^(N+5)));
  Vec(x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3));
};
seq(26)
\\ test: y=Ser(seq(101))*x^6; 0 == x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6
\\ Gheorghe Coserea, Sep 27 2018

Formula

a(n) = b(n-1) + 2*(-1)^n, n >= 4, where b(3)=2, b(n) = (2*(2*n)!/(n!)^2 - (27*n^2+9*n-2)*b(n-1)) / (54*n^2-90*n+32). - Sean A. Irvine, Apr 14 2010

(n - 1)*a(n) = ((3/2)*n - 21/2)*a(n-1) + (8*n - 36)*a(n-2) + ((15/2)*n - 63/2)*a(n-3) + (2*n - 7)*a(n-4). - Simon Plouffe, Feb 09 2012 [Corrected by Matthew House, Sep 03 2024]

Liu Yanpei gives another recurrence. - N. J. A. Sloane, Mar 28 2012

a(n) ~ 2^(2*n+1)/(3^5*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Jul 19 2013

From Gheorghe Coserea, Apr 15 2017: (Start)

G.f.: x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3).

0 = x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6, where y is the g.f. (End)

Extensions

More terms from Sean A. Irvine, Apr 14 2010

Librandi b-file verified by N. J. A. Sloane, Mar 29 2012