A270408 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus g.

5, 93, 1030, 420, 8885, 14065, 65954, 256116, 66066, 442610, 3392843, 3288327, 2762412, 36703824, 85421118, 17454580, 16322085, 344468530, 1558792200, 1171704435, 92400330, 2908358552, 22555934280, 40121261136, 7034538511, 505403910, 22620890127, 276221817810, 945068384880, 600398249550

Offset: 3

Author

Gheorghe Coserea, Mar 17 2016

Keywords

• nonn
• tabf

Comments

Row n contains floor((n-1)/2) terms.

Examples

			Triangle starts:
n\g    [0]          [1]          [2]          [3]          [4]
[3]    5;
[4]    93;
[5]    1030,        420;
[6]    8885,        14065;
[7]    65954,       256116,      66066;
[8]    442610,      3392843,     3288327;
[9]    2762412,     36703824,    85421118,    17454580;
[10]   16322085,    344468530,   1558792200,  1171704435;
[11]   92400330,    2908358552,  22555934280, 40121261136, 7034538511;
[12]   ...
		

Links

• Gheorghe Coserea, Rows n = 3..103, flattened
• Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Crossrefs

Cf. A000365 (column 0).

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) ((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3) (2n-2) (2n-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] (2k-1) (2l-1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
T[n_, g_] := Q[n, 4, g];
Table[T[n, g], {n, 3, 12}, {g, 0, Quotient[n-1, 2]-1}] // Flatten (* Jean-François Alcover, Oct 18 2018 *)

PARI

N = 11; F = 4; gmax(n) = n\2;
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x + O('x^(F+1)));
concat(vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F))))