A058861 Number of 3-connected rooted cubic planar maps with n faces and girth at least 4.

0, 0, 1, 3, 12, 59, 313, 1713, 9559, 54189, 311460, 1812281, 10661303, 63336873, 379601353, 2293205687, 13953099573, 85451824382, 526431271347, 3260689089300, 20296848348929, 126918850161182, 796981464813540

Offset

4,4

Comments

Number of 3-connected triangle-free rooted cubic maps with n faces.

Links

Gheorghe Coserea, Table of n, a(n) for n = 4..308

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*(1-3*x)*g, where g is defined by (x^3-3*x^2+3*x-1)*g^4 + (4*x^4-12*x^3+9*x^2+2*x-3)*g^3 + (6*x^5-10*x^4-15*x^3+36*x^2-14*x-3)*g^2 + (4*x^6+4*x^5-45*x^4+82*x^3-59*x^2+14*x-1)*g + x^7+5*x^6-8*x^5+x^4=0. - Emeric Deutsch, Nov 30 2005

From Gheorghe Coserea, Jul 15 2018: (Start)

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

0 = (x - 1)^3*y^4 - x^2*(x - 1)^2*(2*x - 3)*(2*x + 1)*(3*x - 1)*y^3 + x^4*(x - 1)*(3*x - 1)^2*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*y^2 - x^6*(3*x - 1)^3*(4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*y + x^12*(3*x - 1)^4*(x^3 + 5*x^2 - 8*x + 1).

0 = x^4*(x - 1)^3*(3*x - 1)^4*(256*x^3 - 512*x^2 + 256*x - 27)*(660*x^5 - 2668*x^4 + 4177*x^3 - 3252*x^2 + 1305*x - 220)*y'''' - 4*x^3*(x - 1)^2*(3*x - 1)^3*(696960*x^10 - 5684352*x^9 + 19870624*x^8 - 39218578*x^7 + 48478923*x^6 - 39311914*x^5 + 21210466*x^4 - 7501069*x^3 + 1650104*x^2 - 201370*x + 10395)*y''' + 4*x^2*(x - 1)*(3*x - 1)^2*(7579440*x^12 - 77500656*x^11 + 341548428*x^10 - 862746936*x^9 + 1396393806*x^8 - 1530275829*x^7 + 1167408906*x^6 - 625929723*x^5 + 234247228*x^4 - 59616890*x^3 + 9784582*x^2 - 931830*x + 38940)*y'' - 24*x*(3*x - 1)*(8523900*x^14 - 105292620*x^13 + 561229815*x^12 - 1731677190*x^11 + 3479254732*x^10 - 4837165728*x^9 + 4815815835*x^8 - 3498631418*x^7 + 1868972298*x^6 - 732379803*x^5 + 207693098*x^4 - 41430916*x^3 + 5510394*x^2 - 438095*x + 15730)*y' + 24*(27442800*x^15 - 372895380*x^14 + 2140330050*x^13 - 7047776880*x^12 + 15074631336*x^11 - 22357962673*x^10 + 23891962029*x^9 - 18825921582*x^8 + 11080006886*x^7 - 4892373579*x^6 + 1614037497*x^5 - 392156906*x^4 + 68130318*x^3 - 8004294*x^2 + 569210*x - 18480)*y.

(End)

Maple

eq:=(x^3-3*x^2+3*x-1)*g^4+(4*x^4-12*x^3+9*x^2+2*x-3)*g^3+(6*x^5-10*x^4-15*x^3+36*x^2-14*x-3)*g^2+(4*x^6+4*x^5-45*x^4+82*x^3-59*x^2+14*x-1)*g+x^7+5*x^6-8*x^5+x^4: g:=sum(A[j]*x^j, j=1..37): for n from 1 to 37 do A[n]:=solve(coeff(expand(eq), x^n)=0) od: C3:=x^2*(1-3*x)*g: C3ser:=series(C3, x=0, 34): seq(coeff(C3ser, x^n), n=6..30); # Emeric Deutsch, Nov 30 2005

Prog

(PARI)
F = x^2*(1 - 3*x)*z;
G = x^12*(x - 1)^3*z^4 + x^8*(x - 1)^2*(2*x - 3)*(2*x + 1)*z^3 + x^4*(x - 1)*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*z^2 + (4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*z + (x^3 + 5*x^2 - 8*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) = concat([0, 0], Vec(subst(F, 'z, 'x^4*Z(N))));
seq(21)
\\ test: y = Ser(seq(303))*'x^4; 0 == (x - 1)^3*y^4 - x^2*(x - 1)^2*(2*x - 3)*(2*x + 1)*(3*x - 1)*y^3 + x^4*(x - 1)*(3*x - 1)^2*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*y^2 - x^6*(3*x - 1)^3*(4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*y + x^12*(3*x - 1)^4*(x^3 + 5*x^2 - 8*x + 1)
\\ Gheorghe Coserea, Jul 15 2018

Crossrefs

Cf. A000260, A058859, A058860.

Sequence in context: A298419 A126959 A181328 * A105668 A192768 A179325

Adjacent sequences: A058858 A058859 A058860 * A058862 A058863 A058864

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