A269923 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 3.

1485, 56628, 56628, 1169740, 2668750, 1169740, 17454580, 66449432, 66449432, 17454580, 211083730, 1171704435, 1955808460, 1171704435, 211083730, 2198596400, 16476937840, 40121261136, 40121261136, 16476937840, 2198596400, 20465052608, 196924458720, 647739636160

Offset

6,1

Comments

Row n contains n-5 terms.

Links

Gheorghe Coserea, Rows n = 6..206, 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\f  [1]          [2]          [3]          [4]          [5]
[6]  1485;
[7]  56628,       56628;
[8]  1169740,     2668750,     1169740;
[9]  17454580,    66449432,    66449432,    17454580;
[10] 211083730,   1171704435,  1955808460,  1171704435,  211083730;
[11] ...

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[Q[n, f, 3], {n, 6, 12}, {f, 1, n-5}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

Prog

(PARI)
N = 12; G = 3; gmax(n) = min(n\2, G);
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);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

Crossrefs

Columns f=1-10 give: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.

Row sums give A104742 (column 3 of A269919).

Cf. A269921, A269922, A269924, A269925.

Sequence in context: A045008 A327880 A257713 * A288075 A104742 A278853

Adjacent sequences: A269920 A269921 A269922 * A269924 A269925 A269926

Keyword

nonn,tabl

Author

Gheorghe Coserea, Mar 15 2016

Status

approved