A269922 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 2.

21, 483, 483, 6468, 15018, 6468, 66066, 258972, 258972, 66066, 570570, 3288327, 5554188, 3288327, 570570, 4390386, 34374186, 85421118, 85421118, 34374186, 4390386, 31039008, 313530000, 1059255456, 1558792200, 1059255456, 313530000, 31039008

Offset

4,1

Comments

Row n contains n-3 terms.

Links

Gheorghe Coserea, Rows n = 4..204, 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]
[4]  21;
[5]  483,       483;
[6]  6468,      15018,     6468;
[7]  66066,     258972,    258972,    66066;
[8]  570570,    3288327,   5554188,   3288327,   570570;
[9]  4390386,   34374186,  85421118,  85421118,  34374186,  4390386;
[10] ...

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, 2], {n, 4, 10}, {f, 1, n-3}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

Prog

(PARI)
N = 10; G = 2; 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: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.

Row sums give A006301 (column 2 of A269919).

Cf. A006299 (row maxima), A269921.

Sequence in context: A307600 A025603 A296586 * A006298 A089907 A015695

Adjacent sequences: A269919 A269920 A269921 * A269923 A269924 A269925

Keyword

nonn,tabl

Author

Gheorghe Coserea, Mar 15 2016

Status

approved