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A000256 Number of simple triangulations of plane with n nodes.
(Formerly M2923 N1173)
5
1, 1, 0, 1, 3, 12, 52, 241, 1173, 5929, 30880, 164796, 897380, 4970296, 27930828, 158935761, 914325657, 5310702819, 31110146416, 183634501753, 1091371140915, 6526333259312, 39246152584304, 237214507388796, 1440503185260748 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,5

REFERENCES

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, The enumerative theory of planar maps, pp. 437-448 of J. N. Srivastava, ed., A Survey of Combinatorial Theory, North-Holland, 1973.

LINKS

T. D. Noe, Table of n, a(n) for n=3..200

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; arXiv:0912.0072 [math.NT], 2009.

P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138.

William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

William T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

FORMULA

a(n) = (1/4)*(7*binomial(3*n-9, n-4)-(8*n^2-43*n+57)*a(n-1)) / (8*n^2-51*n+81), n>4. - Vladeta Jovovic, Aug 19 2004

(1/4 + 7/8*n - 9/8*n^3)*a(n) + (-5/4 + 2/3*n + 59/12*n^2 - 13/3*n^3)*a(n+1) + (-1 - 2/3*n + n^2 + 2/3*n^3)*a(n+2). - Simon Plouffe, Feb 09 2012

a(n) ~ 3^(3*n-6+1/2)/(2^(2*n+3)*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Aug 13 2013

From Gheorghe Coserea, Jul 31 2017: (Start)

G.f. y(x) satisfies (with offset 0):

y(x*g^2) = 2 - 1/g, where g=A000260(x). (eqn 2.6 in Tutte's paper)

0 = x*(x+4)^2*y^3 - x*(6*x^2+51*x+76)*y^2 + (12*x^3+108*x^2+115*x-1)*y - (8*x^3+76*x^2+54*x-1).

0 = x*(27*x-4)*deriv(y,x) + x*(7*x+28)*y^2 - 2*(14*x^2+45*x+1)*y + 2*(14*x^2+34*x+1).

(End)

MAPLE

R := RootOf(x-t*(t-1)^2, t); ogf := series( (2*R^3+2*R^2-2*R-1)/((R-1)*(R+1)^2), x=0, 20); # Mark van Hoeij, Nov 08 2011

MATHEMATICA

r = Root[x - t*(t - 1)^2, t, 1] ; CoefficientList[ Series[(2*r^3 + 2*r^2 - 2*r - 1)/((r - 1)*(r + 1)^2), {x, 0, 24}], x] (* Jean-François Alcover, Mar 14 2012, after Maple *)

PROG

(PARI)

A000260_ser(N) = {

  my(v = vector(N, n, binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))));

  Ser(concat(1, v));

};

A000256_seq(N) = {

  my(g = A000260_ser(N)); Vec(subst(2 - 1/g, 'x, serreverse(x*g^2)));

};

A000256_seq(24)

\\ test: y = Ser(A000256_seq(200)); 0 == x*(x+4)^2*y^3 - x*(6*x^2+51*x+76)*y^2 + (12*x^3+108*x^2+115*x-1)*y - (8*x^3+76*x^2+54*x-1)

\\ Gheorghe Coserea, Jul 31 2017

CROSSREFS

First row of array in A210664.

Sequence in context: A010736 A151197 A007198 * A274396 A124202 A138269

Adjacent sequences:  A000253 A000254 A000255 * A000257 A000258 A000259

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Aug 19 2004

STATUS

approved

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Last modified November 19 07:16 EST 2017. Contains 294916 sequences.