A269920 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 0.

1, 1, 1, 2, 5, 2, 5, 22, 22, 5, 14, 93, 164, 93, 14, 42, 386, 1030, 1030, 386, 42, 132, 1586, 5868, 8885, 5868, 1586, 132, 429, 6476, 31388, 65954, 65954, 31388, 6476, 429, 1430, 26333, 160648, 442610, 614404, 442610, 160648, 26333, 1430

Offset

0,4

Comments

Row n contains n+1 terms.

Links

Gheorghe Coserea, Rows n = 0..200, 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]     [7]     [8]
[0]    1;
[1]    1,      1;
[2]    2,      5,      2;
[3]    5,      22,     22,     5;
[4]    14,     93,     164,    93,     14;
[5]    42,     386,    1030,   1030,   386,    42;
[6]    132,    1586,   5868,   8885,   5868,   1586,   132;
[7]    429,    6476,   31388,  65954,  65954,  31388,  6476,   429;
[8]    ...

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, 0], {n, 0, 8}, {f, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

Prog

(PARI)
N = 8; G = 0; 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 k=1-6 give: A000108, A000346, A000184, A000365, A000473, A000502.

Row sums give A000168 (column 0 of A269919).

Cf. A006294 (row maxima).

Sequence in context: A089122 A321577 A268789 * A240706 A240642 A240774

Adjacent sequences: A269917 A269918 A269919 * A269921 A269922 A269923

Keyword

nonn,tabl

Author

Gheorghe Coserea, Mar 14 2016

Status

approved