A058859 Number of 1-connected rooted cubic planar maps with n faces.

1, 3, 19, 143, 1089, 8564, 69075, 569469, 4783377, 40829748, 353395155, 3096104105, 27415923905, 245069538465, 2209155012387, 20064713628389, 183478258249569, 1688112897834496, 15618577076864579, 145242456429736935

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^4*(1-2*x-4*x^2)*m-2*x^8*m^2, where m is defined by 16*x^11*m^4 + (-24*x^9+32*x^8+72*x^7)*m^3 + (-15*x^7-108*x^6-194*x^5-92*x^4+x^3)*m^2 + (-2*x^5-33*x^4-70*x^3-46*x^2+16*x-1)*m - x^2-11*x+1=0. - Emeric Deutsch, Nov 30 2005

From Gheorghe Coserea, Jul 16 2018: (Start):

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

0 = 64*y^4 + (912*x^4 + 640*x^3 + 384*x^2 + 3328*x + 2864)*y^3 - (1743*x^8 + 13968*x^7 + 13344*x^6 - 52888*x^5 - 116934*x^4 - 71248*x^3 - 4064*x^2 + 3768*x - 41)*y^2 + (784*x^12 + 13524*x^11 + 29478*x^10 - 51033*x^9 - 194686*x^8 - 166400*x^7 - 5454*x^6 + 43746*x^5 + 4030*x^4 - 5652*x^3 + 904*x^2 - 41*x)*y - x^5*(x^2 + 11*x - 1)*(1568*x^8 + 476*x^7 - 7456*x^6 - 8458*x^5 - 27*x^4 + 2672*x^3 + 130*x^2 - 330*x + 41).

0 = x*(4*x^2 + 8*x + 5)*(27*x^6 + 216*x^5 + 171*x^4 - 208*x^3 - 339*x^2 + 24*x + 1)*(53687232*x^17 + 962429472*x^16 + 4910442696*x^15 + 11262716564*x^14 + 13535708340*x^13 + 6699339314*x^12 - 8161216832*x^11 - 27707772057*x^10 - 38282906893*x^9 - 23841839272*x^8 + 3164178022*x^7 + 13551725887*x^6 + 6618789645*x^5 + 110368160*x^4 - 189595230*x^3 + 52114000*x^2 - 2282040*x - 80000)*y'''' - (23192884224*x^25 + 642325749120*x^24 + 7010404371072*x^23 + 38396140051536*x^22 + 119087871158520*x^21 + 209055666121344*x^20 + 149537518315396*x^19 - 179206877652920*x^18 - 594068689834972*x^17 - 713069283397760*x^16 - 388115755832091*x^15 + 185412410945637*x^14 + 709124462066474*x^13 + 898548947063912*x^12 + 629038710881040*x^11 + 159866881148998*x^10 - 107640739893374*x^9 - 101244290972424*x^8 - 23418947186993*x^7 + 3644481830365*x^6 + 957436398080*x^5 - 94641974160*x^4 + 1607421440*x^3 + 430075760*x^2 - 17060400*x - 400000)*y''' + (69578652672*x^24 + 1910859372288*x^23 + 21034975582656*x^22 + 114742977687936*x^21 + 350375920009560*x^20 + 585065268522672*x^19 + 317856584972580*x^18 - 736872920930424*x^17 - 1812132349221252*x^16 - 1696870248263700*x^15 - 376785528937023*x^14 + 1026609868750112*x^13 + 1799851001684942*x^12 + 1902275760186412*x^11 + 1364464778889680*x^10 + 504031822062384*x^9 - 75374914747162*x^8 - 173636873122824*x^7 - 67965626046313*x^6 - 3235617436480*x^5 + 1670710238920*x^4 - 60241392600*x^3 - 9066655340*x^2 + 1117875760*x + 15179600)*y'' - 12*(11596442112*x^23 + 315790249536*x^22 + 3414867276384*x^21 + 17899179378120*x^20 + 51714502467480*x^19 + 77928289056012*x^18 + 22675972179932*x^17 - 134244171463804*x^16 - 254323096657040*x^15 - 181481980531415*x^14 + 24427607774667*x^13 + 176309477492908*x^12 + 214672437288248*x^11 + 192416432064275*x^10 + 135698454441595*x^9 + 59484339948854*x^8 + 1838501691038*x^7 - 16090673029130*x^6 - 8704257466200*x^5 - 1085436408240*x^4 + 33590844600*x^3 - 6624333760*x^2 - 719889600*x - 8800000)*y' + 12*(11596442112*x^22 + 313103937024*x^21 + 3232316223360*x^20 + 15530584062240*x^19 + 39522162905640*x^18 + 45540724655832*x^17 - 16695945361396*x^16 - 123726467878420*x^15 - 152050336659260*x^14 - 49261893247550*x^13 + 73707236060447*x^12 + 119787972312984*x^11 + 115583117491500*x^10 + 95686381642950*x^9 + 56811985465335*x^8 + 13932882885644*x^7 - 9032398496482*x^6 - 8810946218840*x^5 - 1354608403560*x^4 + 47155824160*x^3 - 6777547760*x^2 - 855133760*x - 10609600)*y.

(End)

Maple

eq:=16*x^11*m^4+(-24*x^9+32*x^8+72*x^7)*m^3+(-15*x^7-108*x^6-194*x^5-92*x^4+x^3)*m^2+(-2*x^5-33*x^4-70*x^3-46*x^2+16*x-1)*m-x^2-11*x+1: m:=sum(A[j]*x^j, j=0..35): A[0]:=solve(subs(x=0, expand(eq))): for n from 1 to 35 do A[n]:=solve(coeff(expand(eq), x^n)=0) od: C:=(1-2*x-4*x^2)*x^4*m-2*x^8*m^2: Cser:=series(C, x=0, 30): seq(coeff(Cser, x^n), n=4..26); # Emeric Deutsch, Nov 30 2005

Prog

(PARI)
F = x^4*(1-2*x-4*x^2)*z - 2*x^8*z^2;
G = 16*x^11*z^4 - 8*x^7*(3*x^2 - 4*x - 9)*z^3 - x^3*(15*x^4 + 108*x^3 + 194*x^2 + 92*x - 1)*z^2 -  (2*x^5 + 33*x^4 + 70*x^3 + 46*x^2 - 16*x + 1)*z - x^2 - 11*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, Z(N)));
seq(20)
\\ test: y = Ser(seq(303))*'x^4; 0 == 64*y^4 + (912*x^4 + 640*x^3 + 384*x^2 + 3328*x + 2864)*y^3 - (1743*x^8 + 13968*x^7 + 13344*x^6 - 52888*x^5 - 116934*x^4 - 71248*x^3 - 4064*x^2 + 3768*x - 41)*y^2 + (784*x^12 + 13524*x^11 + 29478*x^10 - 51033*x^9 - 194686*x^8 - 166400*x^7 - 5454*x^6 + 43746*x^5 + 4030*x^4 - 5652*x^3 + 904*x^2 - 41*x)*y - x^5*(x^2 + 11*x - 1)*(1568*x^8 + 476*x^7 - 7456*x^6 - 8458*x^5 - 27*x^4 + 2672*x^3 + 130*x^2 - 330*x + 41)
\\ Gheorghe Coserea, Jul 15 2018

Crossrefs

Cf. A000260, A058860, A058861.

Sequence in context: A082758 A110525 A331716 * A291964 A333094 A095002

Adjacent sequences: A058856 A058857 A058858 * A058860 A058861 A058862

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