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A058859 Number of 1-connected rooted cubic planar maps with n faces. 3
1, 3, 19, 143, 1089, 8564, 69075, 569469, 4783377, 40829748, 353395155, 3096104105, 27415923905, 245069538465, 2209155012387, 20064713628389, 183478258249569, 1688112897834496, 15618577076864579, 145242456429736935 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,2
LINKS
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
Sequence in context: A082758 A110525 A331716 * A291964 A333094 A095002
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

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)