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