A269473 - OEIS

%I #36 Dec 30 2018 04:13:40

%S 1,28,664,14912,326496,7048192,150820608,3208396800,67968706048,

%T 1435486650368,30246600953856,636154755940352,13360333295173632,

%U 280258138588839936,5873204471357374464,122980760637407232000,2573349967992101142528,53815038103588370907136

%N a(n) is the number of rooted 2n-face triangulations in an orientable surface of genus 1.

%H Vaclav Kotesovec, <a href="/A269473/b269473.txt">Table of n, a(n) for n = 1..750</a>

%H I. P. Goulden and D. M. Jackson, <a href="https://www.math.uwaterloo.ca/~ipgoulde/KPpapermar08.pdf">The KP hierarchy, branched covers, and triangulations</a>, Advances in Mathematics, Volume 219, Issue 3, 20 October 2008, Pages 932-951.

%H Evgeniy Krasko, Alexander 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 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, Volume 342, Issue 2, February 2019, Pages 584-599.

%F Recurrence: (n-1)*n*(15*n - 46)*a(n) = 48*(270*n^3 - 1503*n^2 + 2478*n - 1280)*a(n-2) - 20736*(3*n - 10)*(3*n - 8)*(15*n - 16)*a(n-4).

%F a(n) ~ 2^(2*n-3) * 3^(3*n/2).

%F From _Gheorghe Coserea_, Jul 27 2018: (Start)

%F a(n+1) = 4^n * n!!/(n+1)! * Sum_{k=0..n} 3^k*(3*n-2*k+1)!!/(n-k)!. (see Krasko link)

%F G.f. y(x) satisfies:

%F 0 = 2*(432*x^2 - 1)^2*y^3 + (432*x^2 - 1)*y^2 + 54*x^2*y + x^2.

%F 0 = x*(432*x^2 - 1)*(108*x^2 + 1)*deriv(y,x) + 2*(432*x^2 - 1)*(648*x^2 + 1)*y^2 + (31104*x^4 + 1116*x^2 + 1)*y + 30*x^2.

%F 0 = (5184*x^2 - 7)*(432*x^2 - 1)^2*y''' + 1296*x*(432*x^2 - 1)*(12096*x^2 - 13)*y'' + 48*(199314432*x^4 - 479088*x^2 + 581)*y' + 663552*x*(2592*x^2 - 11)*y.

%F (End)

%t Rest[CoefficientList[InverseSeries[Series[Sqrt[1728 - 432/x + (30*Sqrt[1 + 48*x] - 54)/x^2 + (Sqrt[1 + 48*x] - 1)/x^3]/864, {x, 0, 20}], x], x]] (* _Vaclav Kotesovec_, Jul 28 2018 *)

%o (PARI)

%o factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);

%o a(n) = {

%o   my(f2 = factorial2);

%o   4^(n-1)*f2(n-1)/n! * sum(k=0, n-1, 3^k * f2(3*n-2*k-2)/(n-1-k)!);

%o };

%o \\ test: y='x*Ser(vector(303, n, a(n))); 0 == 2*(432*x^2 - 1)^2*y^3 + (432*x^2 - 1)*y^2 + 54*x^2*y + x^2

%o \\ _Gheorghe Coserea_, Jul 27 2018

%Y Column k=1 of A266240.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Feb 27 2016