A318102 -id:A318102 - cp's OEIS frontend

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

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

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)

cp's OEIS Frontend

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

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