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A298358 a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.

%I A298358 #47 Sep 24 2018 17:45:39
%S A298358 1,0,0,1,0,3,7,15,63,168,561,1881,6110,21087,72174,250775,883116,
%T A298358 3125910,11174280,40209852,145590720,530358095,1941862860,7144623447,
%U A298358 26403493545,97971775008,364903633215,1363847131450,5113975285788,19233646581282
%N A298358 a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.
%H A298358 Gheorghe Coserea, <a href="/A298358/b298358.txt">Table of n, a(n) for n = 1..501</a>
%H A298358 Daniel Birmajer, Juan B. Gil, Michael D. Weiner, <a href="https://arxiv.org/abs/1803.07727">A family of Bell transformations</a>, arXiv:1803.07727 [math.CO], 2018.
%H A298358 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 A298358 W. T. Tutte, <a href="https://dx.doi.org/10.4153/CJM-1963-029-x">A census of planar maps</a>, Canad. J. Math., 15(1963), 249-271.
%F A298358 G.f.: A(x) = G(x/(1+A(x))^3)-1 where A(x*(G(x))^3) = G(x)-1 and G(x) = g.f. for A000257.
%F A298358 From _Gheorghe Coserea_, Apr 16 2018: (Start)
%F A298358 y = A(x)/x satisfies: 0 = x^6*y^7 + 6*x^5*y^6 + 15*x^4*y^5 + 4*x^3*(5 - 3*x)*y^4 + x^2*(15 - 37*x)*y^3 + x*(16*x^2 - 39*x + 6)*y^2 + (24*x^2 - 15*x + 1)*y + (9*x - 1).
%F A298358 A(x) = serreverse((1+x)^3*(1 + 12*x - (1-4*x)^(3/2))/(2*(4*x+3)^2)); equivalently, it can be rewritten as A(x) = serreverse((y-1)*(y^2+y-1)^3/(y^5*(3*y-2)^2)), where y = A000108(x). (End)
%F A298358 a(n) ~ 3 * 2^(9*n-1) / (sqrt(Pi) * 17^(5/2) * 5^(3*n - 5/2) * n^(5/2)). - _Gheorghe Coserea_ and _Vaclav Kotesovec_, Apr 16 2018
%e A298358 A(x) = x + x^4 + 3*x^6 + 7*x^7 + 15*x^8 + 63*x^9 + 168*x^10 + 561*x^11 + ...
%t A298358 kmax = 30; b[0] = 1; b[n_] := 3*2^(n - 1)*CatalanNumber[n]/(n + 2);
%t A298358 G[x_] = Sum[b[k] x^k, {k, 0, kmax}];
%t A298358 A[_] = 1;
%t A298358 Do[A[x_] = G[x/(1 + A[x] + O[x]^k)^3] - 1 // Normal, {k, 1, kmax + 1}];
%t A298358 CoefficientList[A[x], x][[2 ;; -2]] (* _Jean-François Alcover_, Jun 19 2018 *)
%o A298358 (PARI)
%o A298358 seq(N) = {
%o A298358   my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x), g=(1+4*y-y^2)/4);
%o A298358   Vec(subst(g-1, 'x, serreverse(x*g^3)));
%o A298358 };
%o A298358 seq(30) \\ _Gheorghe Coserea_, Apr 11 2018
%Y A298358 Cf. A000257, A069728, A069729.
%K A298358 nonn
%O A298358 1,6
%A A298358 _Michael D. Weiner_, Jan 17 2018