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%I #33 Sep 08 2022 08:46:23
%S 0,1,15,198,2511,31266,385398,4721004,57590271,700465482,8501284530,
%T 103007201364,1246500179910,15068548264212,182007001727244,
%U 2196875784339288,26501619841355871,319541469851970522,3851239987536347034,46399926869155488708,558853144337650364226
%N a(0)=0; for n>0, a(n) is the number of rooted 4-regular maps on the torus having n vertices.
%H Evgeniy Krasko, Alexander 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 (2019) Vol. 342, Issue 2, 584-599. Also <a href="https://arxiv.org/abs/1709.03225">arXiv</a>, arXiv:1709.03225 [math.CO], 2017. See Theorem 2.1 and Table 1.
%F a(n) = 6^(n-1)*(2^n - (2*n-1)!!/n!) for n>0.
%F G.f.: (1/6)*(1/(1-12*x)-1/sqrt(1-12*x)).
%F D-finite with recurrence: n*a(n) +6*(-4*n+3)*a(n-1) +72*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Jan 09 2020
%t CoefficientList[Series[(1/6) (1/(1 - 12 x) - 1/Sqrt[1 - 12 x]), {x, 0, 33}], x] (* _Vincenzo Librandi_, Jan 10 2020 *)
%o (Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [ 6^(n-1)*(2^n -(DoubleFactorial(2*n-1))/Factorial(n)): n in [0..28] ]; // _Vincenzo Librandi_, Jan 10 2020
%K nonn
%O 0,3
%A _Evgeniy Krasko_, Dec 30 2018
%E Added initial 0 to match generating function and Taylor series in Theorem 2.1. - _N. J. A. Sloane_, Jan 11 2019
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