A270411 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus g.

132, 6476, 160648, 60060, 2762412, 3944928, 37460376, 129726760, 31039008, 429166584, 2908358552, 2583699888, 4331674512, 50534154408, 106853266632, 20465052608, 39599553708, 729734918432, 2979641557620, 2079913241120

Offset: 6

Author

Gheorghe Coserea, Mar 17 2016

Keywords

• nonn
• tabf

Comments

Row n contains floor((n-4)/2) terms.

Examples

			Triangle starts:
n\g    [0]              [1]              [2]              [3]
[6]    132;
[7]    6476;
[8]    160648,          60060;
[9]    2762412,         3944928;
[10]   37460376,        129726760,       31039008;
[11]   429166584,       2908358552,      2583699888;
[12]   4331674512,      50534154408,     106853266632,    20465052608;
[13]   39599553708,     729734918432,    2979641557620,   2079913241120;
[14]   ...
		

Links

• Gheorghe Coserea, Rows n = 6..106, flattened
• Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Crossrefs

Cf. A270410.

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, 7, g];
Table[T[n, g], {n, 6, 13}, {g, 0, Quotient[n-4, 2]-1}] // Flatten

PARI

N = 13; F = 7; 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)));
v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));
concat(v)