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A318102
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Number of rooted 2-connected 4-regular maps on the projective plane, which may have loops, with n inner faces.
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3
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5, 38, 199, 1466, 12365, 109700, 1003929, 9404402, 89690920, 867506788, 8486154214, 83790178300, 833805753167, 8352569222312, 84150924820499, 852039732062530, 8664839058268872, 88459350543053228, 906208005777385526, 9312350891307447116, 95963703215086597466, 991421114632619679480
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OFFSET
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1,1
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LINKS
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FORMULA
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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).
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EXAMPLE
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A(x) = 5*x + 38*x^2 + 199*x^3 + 1466*x^4 + 12365*x^5 + 109700*x^6 + ...
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PROG
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(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))
*/
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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