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

Author

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

Keywords

• nonn

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.

Crossrefs

Cf. A000260, A058859, A058860.

Programs

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

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

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)

Extensions

More terms from Emeric Deutsch, Nov 30 2005