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A000256 Number of simple triangulations of the plane with n nodes.

%I A000256 M2923 N1173 #89 Apr 13 2022 13:25:14
%S A000256 1,1,0,1,3,12,52,241,1173,5929,30880,164796,897380,4970296,27930828,
%T A000256 158935761,914325657,5310702819,31110146416,183634501753,
%U A000256 1091371140915,6526333259312,39246152584304,237214507388796,1440503185260748
%N A000256 Number of simple triangulations of the plane with n nodes.
%C A000256 A triangulation is simple if it contains no separating 3-cycle. The triangulations are rooted with three fixed exterior nodes. - _Andrew Howroyd_, Feb 24 2021
%D A000256 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000256 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000256 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.
%H A000256 T. D. Noe, <a href="/A000256/b000256.txt">Table of n, a(n) for n=3..200</a>
%H A000256 Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, <a href="https://doi.org/10.4230/LIPIcs.AofA.2018.29">Asymptotic Expansions for Sub-Critical Lagrangean Forms</a>, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
%H A000256 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000256 Simon Plouffe, <a href="https://arxiv.org/abs/0912.0072">Une méthode pour obtenir la fonction génératrice d'une série</a>. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.
%H A000256 P. N. Rathie, <a href="http://dx.doi.org/10.1016/0095-8956(74)90055-0">A census of simple planar triangulations</a>, J. Combin. Theory, B 16 (1974), 134-138.
%H A000256 William T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1962-002-9">A census of planar triangulations</a>, Canad. J. Math. 14 (1962), 21-38.
%H A000256 William T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1963-029-x">A Census of Planar Maps</a>, Canad. J. Math. 15 (1963), 249-271.
%F A000256 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
%F A000256 (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
%F A000256 a(n) ~ 3^(3*n-6+1/2)/(2^(2*n+3)*sqrt(Pi)*n^(5/2)). - _Vaclav Kotesovec_, Aug 13 2013
%F A000256 From _Gheorghe Coserea_, Jul 31 2017: (Start)
%F A000256 G.f. y(x) satisfies (with offset 0):
%F A000256 y(x*g^2) = 2 - 1/g, where g=A000260(x). (eqn 2.6 in Tutte's paper)
%F A000256 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).
%F A000256 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).
%F A000256 (End)
%p A000256 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
%t A000256 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 *)
%o A000256 (PARI)
%o A000256 A000260_ser(N) = {
%o A000256   my(v = vector(N, n, binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))));
%o A000256   Ser(concat(1,v));
%o A000256 };
%o A000256 A000256_seq(N) = {
%o A000256   my(g = A000260_ser(N)); Vec(subst(2 - 1/g, 'x, serreverse(x*g^2)));
%o A000256 };
%o A000256 A000256_seq(24)
%o A000256 \\ 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)
%o A000256 \\ _Gheorghe Coserea_, Jul 31 2017
%o A000256 (PARI) seq(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); Vec(2 - sqrt(serreverse( x*(2-g)^2*g^4)/x ))} \\ _Andrew Howroyd_, Feb 23 2021
%Y A000256 First row of array in A210664.
%K A000256 nonn,nice
%O A000256 3,5
%A A000256 _N. J. A. Sloane_
%E A000256 More terms from _Vladeta Jovovic_, Aug 19 2004