OFFSET
1,1
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..301
Stavros Garoufalidis, Marcos Marino, Universality and asymptotics of graph counting problems in nonorientable surfaces, arXiv:0812.1195 [math.CO], 2008-2009.
E. Krasko, A. Omelchenko, Enumeration of r-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps, arXiv:1709.03225 [math.CO], 2017.
E. Krasko, A. Omelchenko, 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, Discrete Mathematics, Volume 342, Issue 2, February 2019, Pages 584-599.
FORMULA
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)
G.f. y=A(x) satisfies:
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).
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.
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.
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
MATHEMATICA
(-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 *)
PROG
(PARI)
seq(N) = {
my(x='x + O('x^(N+2)), c=(1-sqrt(1-12*x))/(6*x));
Vec((1 - x - 3*x*c - sqrt(1 - 4*x - 4*x*c))/x);
};
seq(20)
\\ 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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Jul 08 2018
STATUS
approved