A058860 -id:A058860 - cp's OEIS frontend

A099553 Number of rooted 2-connected loopless 4-regular planar maps with n inner faces.

1, 2, 10, 42, 209, 1066, 5726, 31688, 180234, 1047356, 6198500, 37253790, 226891665, 1397880330, 8699804598, 54629525808, 345778883678, 2204263514460, 14142192816908, 91263177339092, 592069697914170, 3859674384409668, 25272938482712044

Offset: 3

N. J. A. Sloane, Nov 18 2004

nonn

a(n) is also the number of rooted loopless planar maps with n-1 edges and no isthmuses. - Andrew Howroyd, Apr 01 2021

a(n) is also the number of rooted 2-connected plane quadrangulations with n+1 vertices (allowing multiple edges). - Brendan McKay, Apr 08 2025

			A(x) = x^3 + 2*x^4 + 10*x^5 + 42*x^6 + 209*x^7 + 1066*x^8 + 5726*x^9 + ...

Gheorghe Coserea, Table of n, a(n) for n = 3..305
Han Ren, Yanpei Liu and Zhaoxiang Li, Enumeration of 2-connected Loopless 4-regular Maps on the Plane, European J. Combin., 23 (2002), 93-111.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq (7).

Row sums of A342980.

Cf. A000139, A058860, A290326.

A099553 := proc(n)
    local e;
    e := n-1 ;
    add(binomial(2*e-r,e-2-2*r)*2^r*binomial(2*e,r),r=0..floor(e/2-1)) ;
    %-3*add(binomial(2*e-r,e-3-2*r)*2^r*binomial(2*e,r),r=0..floor((e-3)/2)) ;
    %*2/e ;
end proc:
seq(A099553(n),n=3..30) ; # R. J. Mathar, Aug 28 2018

a[n_] := Module[{e, s}, e = n-1; s = Sum[Binomial[2e-r, e-2-2r]*2^r*Binomial[2e, r], {r, 0, Floor[e/2-1]}]; s = s-3*Sum[Binomial[2e-r, e-3-2r]*2^r*Binomial[2e, r], {r, 0, Floor[(e-3)/2]}]; s=2s/e];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Feb 14 2023, after R. J. Mathar *)

F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z);
G = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2;
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+3)));
seq(23)
\\ test: y = Ser(seq(303))*'x^3; 0 == 8*y^4 + (32*x + 12)*y^3 + (48*x^2 + 23*x + 6)*y^2 + (32*x^3 + 10*x^2 - 10*x + 1)*y + x^3*(8*x - 1)
\\ Gheorghe Coserea, Jul 13 2018

seq(n)={my(g=1+x*sum(n=1,n,x^n*binomial(3*n, n)*2/((n+1)*(2*n+1))) + O(x*x^n)); Vec(-1 + sqrt(serreverse(x/g^2)/x))} \\ Andrew Howroyd, Apr 06 2021

From Gheorghe Coserea, Jul 12 2018: (Start)
G.f. A(x) = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z), where z = 1 + 2*x^2 + 6*x^3 + 34*x^4 + 176*x^5 + 1004*x^6 + ... satisfies 0 = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2. (See Theorem D in reference.)
G.f. y=A(x) satisfies:
0 = 8*y^4 + (32*x + 12)*y^3 + (48*x^2 + 23*x + 6)*y^2 + (32*x^3 + 10*x^2 - 10*x + 1)*y + x^3*(8*x - 1).
0 = x^3*(2*x + 1)*(49*x - 18)*(196*x - 27)*y'''' + x^2*(96040*x^3 - 27587*x^2 - 9297*x + 972)*y''' + (72030*x^4 - 36309*x^3 + 2010*x^2 - 864*x)*y'' - 6*(8*x + 3)*(49*x + 12)*y' + (2352*x + 576)*y.
(End)
Conjecture: 3*n *(3*n-1) *(5*n-8) *(3*n-2)*a(n) -(n-2) *(2*n-3) *(355*n^2 -703*n +300)*a(n-1) -98*(n-2) *(5*n-3) *(2*n-3) *(2*n-5) *a(n-2)=0. - R. J. Mathar, Aug 28 2018
G.f.: x*(A(x) - 1) where A(x) satisfies A(x) = G(x*A(x)^2) and (G(x) + 2*x - 1)/x is the g.f. of A000139. - Andrew Howroyd, Apr 06 2021

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

Original entry on oeis.org

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

Offset: 4

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

nonn

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.

Cf. A000260, A058860, A058861.

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

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

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)

More terms from Emeric Deutsch, Nov 30 2005

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

Original entry on oeis.org

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

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

nonn

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

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.

Cf. A000260, A058859, A058860.

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

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

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)

More terms from Emeric Deutsch, Nov 30 2005

A318101 Number of rooted 2-connected 4-regular planar maps, which may have loops, with n inner faces.

Original entry on oeis.org

2, 9, 30, 154, 986, 6977, 52590, 415678, 3409032, 28787498, 248930292, 2195238596, 19682012382, 178974809121, 1647460326046, 15327261314934, 143942130406288, 1363094805806462, 13004498819335396, 124900418475706476, 1206861624598185332, 11725558427958257690, 114494070652568918380

Offset: 2

Gheorghe Coserea, Aug 16 2018

nonn

			A(x) = 2*x^2 + 9*x^3 + 30*x^4 + 154*x^5 + 986*x^6 + 6977*x^7 + 52590*x^8 + ...

Gheorghe Coserea, Table of n, a(n) for n = 2..307
Han Ren, Yanpei Liu and Zhaoxiang Li, Enumeration of 2-connected Loopless 4-regular Maps on the Plane, European J. Combin., 23 (2002), 93-111.

Cf. A058860, A099553, A290326, A318102, A318103.

F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x));
G = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*z + 2;
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((1 + 2*x)*subst(F, 'z, Z(N+2)));
seq(23)
\\ test: y=Ser(seq(303))*x^2; 0 == 8*y^4 + 4*(2*x + 1)*(2*x + 3)*y^3 - (x - 6)*(2*x + 1)^2*y^2 + (2*x + 1)^3*(18*x^2 - 16*x + 1)*y + x^2*(2*x + 1)^4*(27*x - 2)

G.f.: (1 + 2*x)*F, where F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x)) and z = 1 + 2*x + 6*x^2 + 34*x^3 + 254*x^4 + 2052*x^5 + 17332*x^6 + 151658*x^7 + ... satisfies 0 = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*z + 2. (see Theorem C in link)
G.f. y=A(x) satisfies:
0 = 8*y^4 + 4*(2*x + 1)*(2*x + 3)*y^3 - (x - 6)*(2*x + 1)^2*y^2 + (2*x + 1)^3*(18*x^2 - 16*x + 1)*y + x^2*(2*x + 1)^4*(27*x - 2).
0 = x^3*(2*x + 1)^4*(4*x^2 - 2*x + 1)*(108*x^2 - 304*x + 27)*(128*x^6 - 1504*x^5 + 5864*x^4 - 8282*x^3 + 4381*x^2 - 659*x + 60)*y'''' - x^2*(2*x + 1)^3*(417792*x^10 - 1973504*x^9 - 7891840*x^8 + 53958576*x^7 - 106786208*x^6 + 92663096*x^5 - 38721768*x^4 + 9604075*x^3 - 1447438*x^2 + 141966*x - 4860)*y''' + 3*x*(2*x + 1)^2*(163840*x^12 - 1929216*x^11 + 11348480*x^10 - 47888896*x^9 + 125855008*x^8 - 184580160*x^7 + 158611640*x^6 - 81013580*x^5 + 22892592*x^4 - 3821021*x^3 + 403960*x^2 - 23876*x + 120)*y'' - 12*(2*x + 1)*(163840*x^13 - 1888256*x^12 + 11294208*x^11 - 48430080*x^10 + 125093344*x^9 - 184709184*x^8 + 159190952*x^7 - 80413964*x^6 + 23140740*x^5 - 3792653*x^4 + 391233*x^3 - 28410*x^2 - 199*x + 30)*y' + 24*(163840*x^13 - 1847296*x^12 + 11198976*x^11 - 48855552*x^10 + 124699296*x^9 - 184627968*x^8 + 159583928*x^7 - 80114156*x^6 + 23238984*x^5 - 3787577*x^4 + 385076*x^3 - 30072*x^2 - 292*x + 40)*y.
a(n) ~ c / (sqrt(Pi) * n^(5/2) * r^n), where r = (76 - 7*sqrt(103))/54 and c = sqrt(3278181/(3125*(109592 + 10823*sqrt(103)))). - Vaclav Kotesovec, Aug 25 2018

A318102 Number of rooted 2-connected 4-regular maps on the projective plane, which may have loops, with n inner faces.

Original entry on oeis.org

5, 38, 199, 1466, 12365, 109700, 1003929, 9404402, 89690920, 867506788, 8486154214, 83790178300, 833805753167, 8352569222312, 84150924820499, 852039732062530, 8664839058268872, 88459350543053228, 906208005777385526, 9312350891307447116, 95963703215086597466, 991421114632619679480

Offset: 1

Gheorghe Coserea, Aug 19 2018

nonn

			A(x) = 5*x + 38*x^2 + 199*x^3 + 1466*x^4 + 12365*x^5 + 109700*x^6 + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..303
Shude Long, Han Ren, Counting 2-Connected 4-Regular Maps on the Projective Plane, Volume 21, Issue 2 (2014), Paper #P2.51.
Gheorghe Coserea, Annihilating differential operator

A058860, A099553, A290326, A318101, A318103.

F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x));
G = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*z + 2;
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;
};
f(N) = subst((z + 2*z*F - 1)/(2*z^2*x), 'z, Z(N));
Fp2(N) = {
  my(z=Z(N), f=f(N));
  ((1 - sqrt(1 - 4*z^2*x*f))/(x*z) + z*(z-3)/2 * f^2 + 2*(f - 1))/(2*f - 1);
};
Fp4(N) = (1 + 2*x)*(Fp2(N) - 1) + 3*subst(F, 'z, Z(N+2));
seq(N) = Vec(Fp4(N+1));
seq(22)
/* test:
system("wget https://oeis.org/A318102/a318102.txt");
apply_diffop(p, s) = { \\ apply diffop p (encoded as Pol in Dx) to Ser s
  s=intformal(s);
  sum(n=0, poldegree(p, 'Dx), s=s'; polcoeff(p, n, 'Dx) * s);
};
0 == apply_diffop(read("a318102.txt"), Fp4(1001))
*/

G.f.: (1 + 2*x)*(Fp2 - 1) + 3*F, where Fp2 and F are given by the system of algebraic equations:
0 = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*z + 2,
F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x)),
f = (z + 2*z*F - 1)/(2*z^2*x),
Fp2 = ((1 - sqrt(1 - 4*z^2*x*f))/(x*z) + z*(z-3)/2 * f^2 + 2*(f - 1))/(2*f - 1).
The initial coefficients of the solutions are:
z = 1 + 2*x + 6*x^2 + 34*x^3 + 254*x^4 + 2052*x^5 + 17332*x^6 + ...,
F = 2*x^2 + 5*x^3 + 20*x^4 + 114*x^5 + 758*x^6 + 5461*x^7 + 41668*x^8 + ...,
f = 1 + x + 6*x^2 + 37*x^3 + 262*x^4 + 2050*x^5 + 17064*x^6 + ...,
Fp2 = 1 + 5*x + 22*x^2 + 140*x^3 + 1126*x^4 + 9771*x^5 + 87884*x^6 + ...
(see Facts 2-5 and Theorem B in the link)
G.f. y=A(x) satisfies:
0 = 4096*x^7*(4*x^2 - 2*x + 1)^2*y^8 + 2048*x^6*(4*x^2 - 2*x + 1)^2*(36*x^2 + 16*x - 7)*y^7 + 128*x^5*(4*x^2 - 2*x + 1)*(8320*x^6 + 19648*x^5 - 1076*x^4 - 1804*x^3 + 1907*x^2 - 580*x + 126)*y^6 + 32*x^4*(4*x^2 - 2*x + 1)*(225280*x^7 + 444240*x^6 + 84688*x^5 - 29552*x^4 + 32044*x^3 - 9577*x^2 + 976*x - 280)*y^5 + x^3*(40239104*x^11 - 79837184*x^10 + 295013376*x^9 - 58917488*x^8 + 30598624*x^7 + 31536856*x^6 - 14288200*x^5 + 7449849*x^4 - 1791392*x^3 + 303304*x^2 - 15680*x + 2800)*y^4 + 2*x^2*(272629760*x^13 - 24282112*x^12 - 175736320*x^11 + 322666592*x^10 - 42540704*x^9 + 44400384*x^8 + 30919616*x^7 - 7960626*x^6 + 8259482*x^5 - 2256409*x^4 + 613344*x^3 - 92803*x^2 + 2512*x - 252)*y^3 + x*(4137222144*x^14 + 1879746560*x^13 - 1113429024*x^12 + 1342878720*x^11 + 65189712*x^10 + 10079664*x^9 + 147999470*x^8 - 45142196*x^7 + 25711384*x^6 - 6520084*x^5 + 2042177*x^4 - 392900*x^3 + 48476*x^2 - 1064*x + 49)*y^2 - 2*(1128267776*x^16 - 4335727616*x^15 - 6678567648*x^14 + 1061181280*x^13 - 2785972352*x^12 + 213096160*x^11 - 166061526*x^10 - 112334126*x^9 + 50212017*x^8 - 27194278*x^7 + 7091863*x^6 - 1701882*x^5 + 350358*x^4 - 36314*x^3 + 2951*x^2 - 44*x + 1)*y + x*(17448304640*x^16 - 38432538624*x^15 + 29298729744*x^14 - 1261398240*x^13 + 9372670936*x^12 + 6841726488*x^11 + 1476038993*x^10 + 1644370884*x^9 + 177903076*x^8 + 98892200*x^7 + 15461596*x^6 - 2656592*x^5 + 901090*x^4 - 145464*x^3 + 25339*x^2 - 364*x + 10).

A318103 Number of rooted 2-connected loopless 4-regular maps on the projective plane with n inner faces.

Original entry on oeis.org

6, 21, 138, 781, 4836, 30099, 191698, 1236024, 8063492, 53086930, 352249244, 2352800079, 15805224904, 106702428453, 723509453442, 4924851788720, 33638721268140, 230477992427450, 1583550831926508, 10907729315809642, 75307599054762424, 521026923863915206, 3611800088179535100

Offset: 2

Gheorghe Coserea, Aug 20 2018

nonn

			A(x) = 6*x^2 + 21*x^3 + 138*x^4 + 781*x^5 + 4836*x^6 + 30099*x^7 + ...

Gheorghe Coserea, Table of n, a(n) for n = 2..304
Shude Long, Han Ren, Counting 2-Connected 4-Regular Maps on the Projective Plane, Volume 21, Issue 2 (2014), Paper #P2.51.
Gheorghe Coserea, Annihilating differential operator

Cf. A058860, A099553, A290326, A318101, A318102.

F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z);
G = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2;
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;
};
f(N) = subst((z - 1 + 2*z*x + 2*z*F)/(2*x*z^2), 'z, Z(N));
Fp4(N) = {
  my(z=Z(N), f=f(N));
((1 - sqrt(1- 4*x*z^2*f))/(x*z) - z*f - x*(z*f)^3*(2 - f))/(2*f - 1) - 1;
};
seq(N) = Vec(Fp4(N+2));
seq(23)
/* test:
system("wget https://oeis.org/A318103/a318103.txt");
apply_diffop(p, s) = {
  s=intformal(s);
  sum(n=0, poldegree(p, 'Dx), s=s'; polcoeff(p, n, 'Dx) * s);
};
0 == apply_diffop(read("a318103.txt"), Fp4(1001))
*/

G.f.: ((1 - sqrt(1- 4*x*z^2*f))/(x*z) - z*f - x*(z*f)^3*(2 - f))/(2*f - 1) - 1, where z and f are given by the system of algebraic equations:
0 = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2,
F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z),
f = (z - 1 + 2*z*x + 2*z*F)/(2*x*z^2).
The initial coefficients of the solutions are:
z = 1 + 2*x^2 + 6*x^3 + 34*x^4 + 176*x^5 + 1004*x^6 + 5858*x^7 + ...
F = x^3 + 2*x^4 + 10*x^5 + 42*x^6 + 209*x^7 + 1066*x^8 + 5726*x^9 + ...
f = 1 + x + 2*x^2 + 9*x^3 + 42*x^4 + 222*x^5 + 1232*x^6 + 7137*x^7 + ...
(see Facts 6-7 and Theorem C in the link)
G.f. y=A(x) satisfies:
0 = 4096*x^7*(2*x + 1)^2*y^8 + 2048*x^6*(2*x + 1)^2*(16*x - 7)*y^7 + 128*x^5*(2*x + 1)*(1792*x^3 - 285*x^2 - 76*x + 126)*y^6 + 32*x^4*(2*x + 1)*(14336*x^4 - 2360*x^3 - 57*x^2 - 144*x - 280)*y^5 + x^3*(1146880*x^6 + 625920*x^5 + 282633*x^4 + 174368*x^3 + 44232*x^2 + 6720*x + 2800)*y^4 + 2*x^2*(2*x + 1)*(229376*x^6 + 108288*x^5 + 419113*x^4 + 53390*x^3 - 39619*x^2 + 1000*x - 252)*y^3 + x*(458752*x^8 + 740608*x^7 + 3399862*x^6 + 1371564*x^5 - 317093*x^4 - 58308*x^3 + 25400*x^2 - 672*x + 49)*y^2 + 2*(65536*x^9 + 162048*x^8 + 1258098*x^7 + 287981*x^6 - 86682*x^5 + 22504*x^4 + 5250*x^3 - 2026*x^2 + 36*x - 1)*y + x^2*(16384*x^7 + 58112*x^6 + 674825*x^5 + 33912*x^4 + 11954*x^3 + 23076*x^2 - 390*x + 12).
From Vaclav Kotesovec, Aug 25 2018: (Start)
a(n) ~ c1 * (196/27)^n / n^(5/4) * (1 + c2/n^(1/4) + c3/n^(1/2)), where
c1 = 7^(5/4) * Gamma(1/4) / (5^(5/4) * 3^(3/4) * Pi),
c2 = -17 * 7^(1/4) * sqrt(Pi) / (3^(7/4) * 5^(1/4) * Gamma(1/4)),
c3 = 71 * sqrt(7) * Pi / (2^(3/2) * sqrt(3) * 5^(3/2) * Gamma(1/4)^2). (End)

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A318101 Number of rooted 2-connected 4-regular planar maps, which may have loops, with n inner faces.

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A318102 Number of rooted 2-connected 4-regular maps on the projective plane, which may have loops, with n inner faces.

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A318103 Number of rooted 2-connected loopless 4-regular maps on the projective plane with n inner faces.

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