A288082 a(n) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus 2.

483, 15018, 258972, 3288327, 34374186, 313530000, 2583699888, 19678611645, 140725699686, 955708437684, 6216591472728, 38985279745230, 236923660397172, 1401097546161936, 8089830217844928, 45732525474843801, 253705943922820830, 1383896652090932364, 7434748752218650632, 39394773780853063650

Offset: 5

Author

Gheorghe Coserea, Jun 05 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 2 with n edges and f faces for 1<=f<=10: A006298 f=1, this sequence, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.

Column 2 of A269922, column 2 of A270406.

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, 2, 2];
Table[a[n], {n, 5, 24}] (* Jean-François Alcover, Oct 18 2018 *)

PARI

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288082_ser(N) = {
  my(y = A000108_ser(N+1));
  3*y*(y-1)^5*(7*y^4 + 294*y^3 + 309*y^2 - 547*y + 98)/(y-2)^14;
};
Vec(A288082_ser(20))