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

2,1

Links

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.

Formula

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

Example

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

Prog

(PARI)
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)

Crossrefs

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

Sequence in context: A203365 A181954 A343565 * A358652 A347893 A151823

Adjacent sequences: A318098 A318099 A318100 * A318102 A318103 A318104

Keyword

nonn

Author

Gheorghe Coserea, Aug 16 2018

Status

approved