Voozh

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

6,3

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

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.

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)

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A000256.

Sequence in context: A057343 A377117 A343466 * A032087 A165164 A241602