A288262 a(n) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus 3.

174437377400, 19925913354061, 1115525500250760, 41491242915292306, 1165172136542282424, 26522236056202555206, 511895831411154922176, 8640883781524178188980, 130468023103972196647776, 1792206112041706943912462, 22695416350294243544684240, 267740228837597817351215676

Offset: 13

Author

Gheorghe Coserea, Jun 07 2017

Keywords

• nonn

Links

• Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, this sequence, A288263 f=9, A288264 f=10.

Column 8 of A269923.

Cf. A000108.

Programs

Mathematica

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n + 1) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 Q[n - 2, f, g - 1] + 1/2 Sum[l = n - k; Sum[v = f - u; Sum[j = g - i; Boole[l >= 1 && v >= 1 && j >= 0] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 8, 3];
Table[a[n], {n, 13, 24}] (* Jean-François Alcover, Oct 17 2018 *)

PARI

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288262_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^13*(2675326679856*y^12 + 54684388381464*y^11 + 178122315841075*y^10 - 372236561648447*y^9 - 717438005317146*y^8 + 1482970059363466*y^7 - 17264319660476*y^6 - 1294789702753096*y^5 + 770104389507952*y^4 - 4493523304288*y^3 - 105563098094272*y^2 + 24298454684800*y - 895286303488)/(y-2)^38;
};
Vec(A288262_ser(12))