A270406 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus g.

1, 5, 22, 10, 93, 167, 386, 1720, 483, 1586, 14065, 15018, 6476, 100156, 258972, 56628, 26333, 649950, 3288327, 2668750, 106762, 3944928, 34374186, 66449432, 12317877, 431910, 22764165, 313530000, 1171704435, 792534015, 1744436, 126264820, 2583699888, 16476937840, 26225260226, 4304016990

Offset: 1

Author

Gheorghe Coserea, Mar 16 2016

Keywords

• nonn
• tabf

Comments

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

Examples

			Triangle starts:
n\g    [0]          [1]          [2]          [3]          [4]
[1]    1;
[2]    5;
[3]    22,          10;
[4]    93,          167;
[5]    386,         1720,        483;
[6]    1586,        14065,       15018;
[7]    6476,        100156,      258972,      56628;
[8]    26333,       649950,      3288327,     2668750;
[9]    106762,      3944928,     34374186,    66449432,    12317877;
[10]   431910,      22764165,    313530000,   1171704435,  792534015;
[11]   ...
		

Links

• Gheorghe Coserea, Rows n = 1..101, flattened
• Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Crossrefs

Columns k=0-1 give: A000346, A006295.

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}]);
Table[Table[Q[n, 2, g], {g, 0, (n+1)/2-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

PARI

N = 10; F = 2; 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))))