A298358 a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.
1, 0, 0, 1, 0, 3, 7, 15, 63, 168, 561, 1881, 6110, 21087, 72174, 250775, 883116, 3125910, 11174280, 40209852, 145590720, 530358095, 1941862860, 7144623447, 26403493545, 97971775008, 364903633215, 1363847131450, 5113975285788, 19233646581282
Offset: 1
Keywords
Examples
A(x) = x + x^4 + 3*x^6 + 7*x^7 + 15*x^8 + 63*x^9 + 168*x^10 + 561*x^11 + ...
Links
- Gheorghe Coserea, Table of n, a(n) for n = 1..501
- Daniel Birmajer, Juan B. Gil, Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
- Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
- W. T. Tutte, A census of planar maps, Canad. J. Math., 15(1963), 249-271.
Programs
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Mathematica
kmax = 30; b[0] = 1; b[n_] := 3*2^(n - 1)*CatalanNumber[n]/(n + 2); G[x_] = Sum[b[k] x^k, {k, 0, kmax}]; A[_] = 1; Do[A[x_] = G[x/(1 + A[x] + O[x]^k)^3] - 1 // Normal, {k, 1, kmax + 1}]; CoefficientList[A[x], x][[2 ;; -2]] (* Jean-François Alcover, Jun 19 2018 *) -
PARI
seq(N) = { my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x), g=(1+4*y-y^2)/4); Vec(subst(g-1, 'x, serreverse(x*g^3))); }; seq(30) \\ Gheorghe Coserea, Apr 11 2018
Formula
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.
From Gheorghe Coserea, Apr 16 2018: (Start)
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).
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)
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