A288071 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 1.

420, 14065, 256116, 3392843, 36703824, 344468530, 2908358552, 22620890127, 164767964504, 1137369687454, 7506901051000, 47700234551918, 293370096957504, 1753945289216484, 10229201477344752, 58364244137596695, 326571194881454376, 1795631576981016038, 9718877491130813368, 51858415558095569962

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 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, this sequence, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.

Column 4 of A269921.

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, 4, 1];
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);
A288071_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^5*(307*y^4 + 2411*y^3 - 2094*y^2 - 708*y + 504)/(y-2)^14;
};
Vec(A288071_ser(20))