|
%I #12 Aug 28 2018 03:23:08
%S 6,21,138,781,4836,30099,191698,1236024,8063492,53086930,352249244,
%T 2352800079,15805224904,106702428453,723509453442,4924851788720,
%U 33638721268140,230477992427450,1583550831926508,10907729315809642,75307599054762424,521026923863915206,3611800088179535100
%N Number of rooted 2-connected loopless 4-regular maps on the projective plane with n inner faces.
%H Gheorghe Coserea, <a href="/A318103/b318103.txt">Table of n, a(n) for n = 2..304</a>
%H Shude Long, Han Ren, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p51">Counting 2-Connected 4-Regular Maps on the Projective Plane</a>, Volume 21, Issue 2 (2014), Paper #P2.51.
%H Gheorghe Coserea, <a href="/A318103/a318103.txt">Annihilating differential operator</a>
%F 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:
%F 0 = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2,
%F F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z),
%F f = (z - 1 + 2*z*x + 2*z*F)/(2*x*z^2).
%F The initial coefficients of the solutions are:
%F z = 1 + 2*x^2 + 6*x^3 + 34*x^4 + 176*x^5 + 1004*x^6 + 5858*x^7 + ...
%F F = x^3 + 2*x^4 + 10*x^5 + 42*x^6 + 209*x^7 + 1066*x^8 + 5726*x^9 + ...
%F f = 1 + x + 2*x^2 + 9*x^3 + 42*x^4 + 222*x^5 + 1232*x^6 + 7137*x^7 + ...
%F (see Facts 6-7 and Theorem C in the link)
%F G.f. y=A(x) satisfies:
%F 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).
%F From _Vaclav Kotesovec_, Aug 25 2018: (Start)
%F a(n) ~ c1 * (196/27)^n / n^(5/4) * (1 + c2/n^(1/4) + c3/n^(1/2)), where
%F c1 = 7^(5/4) * Gamma(1/4) / (5^(5/4) * 3^(3/4) * Pi),
%F c2 = -17 * 7^(1/4) * sqrt(Pi) / (3^(7/4) * 5^(1/4) * Gamma(1/4)),
%F c3 = 71 * sqrt(7) * Pi / (2^(3/2) * sqrt(3) * 5^(3/2) * Gamma(1/4)^2). (End)
%e A(x) = 6*x^2 + 21*x^3 + 138*x^4 + 781*x^5 + 4836*x^6 + 30099*x^7 + ...
%o (PARI)
%o F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z);
%o G = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2;
%o Z(N) = {
%o my(z0=1+O('x^N), z1=0, n=1);
%o while (n++,
%o z1 = z0 - subst(G, 'z, z0)/subst(deriv(G, 'z), 'z, z0);
%o if (z1 == z0, break()); z0 = z1);
%o z0;
%o };
%o f(N) = subst((z - 1 + 2*z*x + 2*z*F)/(2*x*z^2), 'z, Z(N));
%o Fp4(N) = {
%o my(z=Z(N), f=f(N));
%o ((1 - sqrt(1- 4*x*z^2*f))/(x*z) - z*f - x*(z*f)^3*(2 - f))/(2*f - 1) - 1;
%o };
%o seq(N) = Vec(Fp4(N+2));
%o seq(23)
%o /* test:
%o system("wget https://oeis.org/A318103/a318103.txt");
%o apply_diffop(p, s) = {
%o s=intformal(s);
%o sum(n=0, poldegree(p, 'Dx), s=s'; polcoeff(p, n, 'Dx) * s);
%o };
%o 0 == apply_diffop(read("a318103.txt"), Fp4(1001))
%o */
%Y Cf. A058860, A099553, A290326, A318101, A318102.
%K nonn
%O 2,1
%A _Gheorghe Coserea_, Aug 20 2018
| 