A322914 a(0)=0; for n>0, a(n) is the number of rooted 4-regular maps on the torus having n vertices.

0, 1, 15, 198, 2511, 31266, 385398, 4721004, 57590271, 700465482, 8501284530, 103007201364, 1246500179910, 15068548264212, 182007001727244, 2196875784339288, 26501619841355871, 319541469851970522, 3851239987536347034, 46399926869155488708, 558853144337650364226

Offset: 0

Author

Evgeniy Krasko, Dec 30 2018

Keywords

• nonn

Links

• Evgeniy Krasko, Alexander 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 (2019) Vol. 342, Issue 2, 584-599. Also arXiv, arXiv:1709.03225 [math.CO], 2017. See Theorem 2.1 and Table 1.

Programs

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

Mathematica

CoefficientList[Series[(1/6) (1/(1 - 12 x) - 1/Sqrt[1 - 12 x]), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 10 2020 *)

Formula

a(n) = 6^(n-1)*(2^n - (2*n-1)!!/n!) for n>0.

G.f.: (1/6)*(1/(1-12*x)-1/sqrt(1-12*x)).

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

Extensions

Added initial 0 to match generating function and Taylor series in Theorem 2.1. - N. J. A. Sloane, Jan 11 2019