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%I M3290 N1326 #88 Apr 13 2022 13:25:14
%S 1,0,4,6,24,66,214,676,2209,7296,24460,82926,284068,981882,3421318,
%T 12007554,42416488,150718770,538421590,1932856590,6969847486,
%U 25237057110,91729488354,334589415276,1224445617889,4494622119424
%N Number of rooted polyhedral graphs with n edges.
%C a(n) appears to be odd if and only if n = 2^k - 2 for some integer k >= 3. - _Lewis Chen_, May 05 2019
%D Handbook of Combinatorics, North-Holland '95, p. 892. (Gives different last term)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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
%H Vincenzo Librandi, <a href="/A000287/b000287.txt">Table of n, a(n) for n = 6..1000</a>
%H A. J. W. Duijvestijn and P. J. Federico, <a href="https://doi.org/10.1090/S0025-5718-1981-0628713-3">The number of polyhedral (3-connected planar) graphs</a>, Math. Comp. 37 (1981), no. 156, 523-532.
%H Hsien-Kuei Hwang, Mihyun Kang, 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 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 Simon Plouffe, <a href="http://arxiv.org/abs/0912.0072"> Une méthode pour obtenir la fonction génératrice d'une série</a>, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
%H W. T. Tutte, <a href="https://doi.org/10.1090/S0002-9904-1962-10793-9">A new branch of enumerative graph theory</a>, Bull. Amer. Math. Soc., 68 (1962), 500-504.
%H W. 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.
%H Liu Yanpei, <a href="https://doi.org/10.1016/0095-8956(84)90018-2">On the number of rooted c-nets</a>, J. Combin. Theory, B 36 (1984), 118-123.
%F 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
%F (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
%F Liu Yanpei gives another recurrence. - _N. J. A. Sloane_, Mar 28 2012
%F a(n) ~ 2^(2*n+1)/(3^5*sqrt(Pi)*n^(5/2)). - _Vaclav Kotesovec_, Jul 19 2013
%F From _Gheorghe Coserea_, Apr 15 2017: (Start)
%F 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).
%F 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)
%e G.f. = x^6 + 4*x^8 + 6*x^9 + 24*x^10 + 66*x^11 + 214*x^12 + 676*x^13 + ...
%t 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 *)
%o (PARI)
%o seq(N) = {
%o my(x='x+O('x^(N+5)));
%o Vec(x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3));
%o };
%o seq(26)
%o \\ 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
%o \\ _Gheorghe Coserea_, Sep 27 2018
%Y Cf. A000256.
%K nonn,nice
%O 6,3
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms from _Sean A. Irvine_, Apr 14 2010
%E Librandi b-file verified by _N. J. A. Sloane_, Mar 29 2012
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