A058860 Number of 2-connected rooted cubic planar maps with n faces.

1, 3, 19, 128, 909, 6737, 51683, 407802, 3293497, 27122967, 227095683, 1928656876, 16582719509, 144125955717, 1264625068163, 11190598332502, 99776445196977, 895685185070155, 8090065969366259, 73480719648381240, 670821169614526749

Offset

4,2

Links

Gheorghe Coserea, Table of n, a(n) for n = 4..306

Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps

Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps, Annals of Combinatorics, 6 (2002), no. 3-4, 313-325.

Formula

G.f.: x^2*(f-x)*(1-2*x)/(1+x), where f is defined by 16*x^2*f^3 + (8*x^4+24*x^3+72*x^2+8*x)*f^2 + (x^6+6*x^5-5*x^4-40*x^3+3*x^2-14*x+1)*f - x^4-3*x^3+13*x^2-x=0. - Emeric Deutsch, Nov 30 2005

From Gheorghe Coserea, Jul 14 2018: (Start)

G.f. y=A(x) satisfies:

0 = 16*y^3 - 8*x*(2*x - 1)*(x^2 + 8*x + 1)*y^2 + x^2*(2*x - 1)^2*(x^4 + 20*x^3 + 50*x^2 - 16*x + 1)*y - x^6*(2*x - 1)^3*(x^2 + 11*x - 1).

0 = x^3*(2*x - 1)^3*(x - 2)*(4*x - 5)*(2*x^2 + 10*x - 1)*y''' - x^2*(2*x - 1)^2*(96*x^5 + 188*x^4 - 1570*x^3 + 1791*x^2 - 481*x + 35)*y'' + 12*x*(2*x - 1)*(48*x^6 + 104*x^5 - 898*x^4 + 1186*x^3 - 514*x^2 + 95*x - 5)*y' - 6*(256*x^7 + 608*x^6 - 5456*x^5 + 8292*x^4 - 4962*x^3 + 1525*x^2 - 220*x + 10)*y.

(End)

Example

G.f. = x^4 + 3*x^5 + 19*x^6 + 128*x^7 + 909*x^8 + 6737*x^9 + 51683*x^10 + ... - Michael Somos, Jul 22 2018

Maple

eq:=16*x^2*f^3+(8*x^4+24*x^3+72*x^2+8*x)*f^2+(x^6+6*x^5-5*x^4-40*x^3+3*x^2-14*x+1)*f-x^4-3*x^3+13*x^2-x: f:=sum(A[j]*x^j, j=1..35): for n from 1 to 35 do A[n]:=solve(coeff(expand(eq), x^n)=0) od: C2:=x^2*(f-x)*(1-2*x)/(1+x): C2ser:=series(C2, x=0, 30): seq(coeff(C2ser, x^n), n=4..26); # Emeric Deutsch, Nov 30 2005

Prog

(PARI)
F = x^2*(z - x)*(1 - 2*x)/(1 + x);
G = 16*x^4*z^3 + x*(8*x^4 + 24*x^3 + 72*x^2 + 8*x)*z^2 + (x^6 + 6*x^5 -5*x^4 -40*x^3 + 3*x^2 - 14*x + 1)*z - x^3 - 3*x^2 + 13*x - 1;
Z(N) = {
  my(z0 = 1 + O('x^N), z1=0, n=1);
  while (n++,
    z1 = z0 - subst(G, 'z, z0)/subst(deriv(G, 'z), 'z, z0);
    if (z1 == z0, break()); z0 = z1); z0;
};
seq(N) = Vec(subst(F, 'z, 'x*Z(N+1)));
seq(21)
\\ test: y=Ser(seq(303), 'x)*x^4; 0 == 16*y^3 - 8*x*(2*x - 1)*(x^2 + 8*x + 1)*y^2 + x^2*(2*x - 1)^2*(x^4 + 20*x^3 + 50*x^2 - 16*x + 1)*y - x^6*(2*x - 1)^3*(x^2 + 11*x - 1)
\\ Gheorghe Coserea, Jul 14 2018

Crossrefs

Cf. A000260, A058859, A058861.

Sequence in context: A027308 A295371 A156069 * A074568 A219053 A074713

Adjacent sequences: A058857 A058858 A058859 * A058861 A058862 A058863

Keyword

nonn

Author

N. J. A. Sloane, Jan 06 2001; revised Feb 17 2006

Extensions

More terms from Emeric Deutsch, Nov 30 2005

Status

approved