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A000287 Number of rooted polyhedral graphs with n edges.
(Formerly M3290 N1326)
4
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

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).

Tutte, W. T. 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.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. 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 + 4) a(n) = (3/2 n - 3) a(n - 1) + (8 n + 4) a(n - 2) + (15/2 n + 6) a(n - 3) + (2 n + 3) a(n - 4). [Simon Plouffe, Feb 09 2012]

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) x='x+O('x^31);

Vec(x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3)) \\ Gheorghe Coserea, Apr 15 2017

CROSSREFS

Cf. A000256.

Sequence in context: A240290 A067001 A057343 * A032087 A165164 A241602

Adjacent sequences:  A000284 A000285 A000286 * A000288 A000289 A000290

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from Sean A. Irvine, Apr 14 2010.

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

STATUS

approved

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Last modified May 24 17:32 EDT 2017. Contains 286997 sequences.