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

6,1

Comments

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

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.

Example

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]   ...

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

Prog

(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)

Crossrefs

Cf. A270410.

Sequence in context: A228333 A035837 A335857 * A258493 A184893 A035818

Adjacent sequences: A270408 A270409 A270410 * A270412 A270413 A270414

Keyword

nonn,tabf

Author

Gheorghe Coserea, Mar 17 2016

Status

approved