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A298358
a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.
2
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
1,6
LINKS
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.
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
EXAMPLE
A(x) = x + x^4 + 3*x^6 + 7*x^7 + 15*x^8 + 63*x^9 + 168*x^10 + 561*x^11 + ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael D. Weiner, Jan 17 2018
STATUS
approved