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%I #28 Oct 04 2019 05:07:51
%S 5,38,331,3098,30330,306276,3163737,33252050,354312946,3817498004,
%T 41510761346,454882507468,5017662052868,55664182358808,
%U 620592559670979,6949200032479746,78117065527443654,881170275583541004,9970663315885385502,113137928354523348300
%N a(n) is the number of rooted quadrangulations of the projective plane with n vertices.
%H Gheorghe Coserea, <a href="/A316598/b316598.txt">Table of n, a(n) for n = 1..301</a>
%H Stavros Garoufalidis, Marcos Marino, <a href="https://arxiv.org/abs/0812.1195">Universality and asymptotics of graph counting problems in nonorientable surfaces</a>, arXiv:0812.1195 [math.CO], 2008-2009.
%H E. Krasko, A. Omelchenko, <a href="https://arxiv.org/abs/1709.03225">Enumeration of r-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps</a>, arXiv:1709.03225 [math.CO], 2017.
%H E. Krasko, A. Omelchenko, <a href="https://doi.org/10.1016/j.disc.2018.07.013">Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus</a>, Discrete Mathematics, Volume 342, Issue 2, February 2019, Pages 584-599.
%F G.f. A(x) = (1-x-3*x*c - sqrt(1-4*x-4*x*c))/x, where c=(1-sqrt(1-12*x))/(6*x). (see eqn. (117) in Garoufalidis link)
%F G.f. y=A(x) satisfies:
%F 0 = 3*x^3*y^4 + 6*x^2*(2*x - 1)*y^3 + x*(18*x^2 + 24*x + 1)*y^2 + 2*(6*x^3 + 33*x^2 + 4*x - 1)*y + x*(3*x^2 + 36*x + 10).
%F 0 = 13*x*(4*x + 1)*(12*x - 1)^3*y''''' + (36864*x^4 + 3840*x^3 + 8832*x^2 + 1556*x - 65)*(12*x - 1)^2*y'''' + 16*(248832*x^4 - 5184*x^3 + 29799*x^2 + 2418*x - 259)*(12*x - 1)*y''' + 72*(1382400*x^4 - 201600*x^3 + 144312*x^2 - 4157*x - 492)*y'' + 144*(276480*x^3 - 51840*x^2 + 31488*x - 979)*y' + 165888*y.
%F 0 = x*(4*x + 1)*(48*x^2 - 6*x + 1)*(12*x - 1)^3*y'''' + 2*(10368*x^4 + 12*x^2 + 47*x - 2)*(12*x - 1)^2*y''' + 6*(86400*x^4 - 10800*x^3 + 2472*x^2 + 132*x - 19)*(12*x - 1)*y'' + (2488320*x^4 - 622080*x^3 + 186192*x^2 - 10728*x - 144)*y' + (10368*x - 648)*y.
%F a(n) ~ 2^(2*n + 1/2) * 3^(n + 1/2)/ (Gamma(3/4) * n^(5/4)) * (1 - sqrt(3) * Gamma(3/4) / (sqrt(2*Pi) * n^(1/4))). - _Vaclav Kotesovec_, Oct 04 2019
%t (-6*x + 3*Sqrt[1-12*x] - 2*Sqrt[-36*x + 6*Sqrt[1-12*x] + 3] + 3)/(6*x^2) + O[x]^20 // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 24 2019 *)
%o (PARI)
%o seq(N) = {
%o my(x='x + O('x^(N+2)), c=(1-sqrt(1-12*x))/(6*x));
%o Vec((1 - x - 3*x*c - sqrt(1 - 4*x - 4*x*c))/x);
%o };
%o seq(20)
%o \\ test: y='x*Ser(seq(300), 'x); 0 == 3*x^3*y^4 + (12*x^3 - 6*x^2)*y^3 + (18*x^3 + 24*x^2 + x)*y^2 + (12*x^3 + 66*x^2 + 8*x - 2)*y + (3*x^3 + 36*x^2 + 10*x)
%Y Cf. A005159, A007137, A278120.
%K nonn
%O 1,1
%A _Gheorghe Coserea_, Jul 08 2018
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