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

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

Offset

1,1

Links

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

Formula

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).

Example

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

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;
};
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))
*/

Crossrefs

A058860, A099553, A290326, A318101, A318103.

Sequence in context: A280437 A222646 A145156 * A163698 A294053 A345686

Adjacent sequences: A318099 A318100 A318101 * A318103 A318104 A318105

Keyword

nonn

Author

Gheorghe Coserea, Aug 19 2018

Status

approved