A269473 a(n) is the number of rooted 2n-face triangulations in an orientable surface of genus 1.

1, 28, 664, 14912, 326496, 7048192, 150820608, 3208396800, 67968706048, 1435486650368, 30246600953856, 636154755940352, 13360333295173632, 280258138588839936, 5873204471357374464, 122980760637407232000, 2573349967992101142528, 53815038103588370907136

Offset: 1

Author

Vaclav Kotesovec, Feb 27 2016

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..750
• I. P. Goulden and D. M. Jackson, The KP hierarchy, branched covers, and triangulations, Advances in Mathematics, Volume 219, Issue 3, 20 October 2008, Pages 932-951.
• Evgeniy Krasko, Alexander Omelchenko, Enumeration of r-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps, arXiv:1709.03225 [math.CO], 2017.
• 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, Volume 342, Issue 2, February 2019, Pages 584-599.

Crossrefs

Column k=1 of A266240.

Programs

Mathematica

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 *)

PARI

factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);
a(n) = {
  my(f2 = factorial2);
  4^(n-1)*f2(n-1)/n! * sum(k=0, n-1, 3^k * f2(3*n-2*k-2)/(n-1-k)!);
};
\\ 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
\\ Gheorghe Coserea, Jul 27 2018

Formula

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).

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

From Gheorghe Coserea, Jul 27 2018: (Start)

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

G.f. y(x) satisfies:

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

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.

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.

(End)