A270406 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus g.

1, 5, 22, 10, 93, 167, 386, 1720, 483, 1586, 14065, 15018, 6476, 100156, 258972, 56628, 26333, 649950, 3288327, 2668750, 106762, 3944928, 34374186, 66449432, 12317877, 431910, 22764165, 313530000, 1171704435, 792534015, 1744436, 126264820, 2583699888, 16476937840, 26225260226, 4304016990

Offset

1,2

Comments

Row n contains floor((n+1)/2) terms.

Links

Gheorghe Coserea, Rows n = 1..101, 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]          [4]
[1]    1;
[2]    5;
[3]    22,          10;
[4]    93,          167;
[5]    386,         1720,        483;
[6]    1586,        14065,       15018;
[7]    6476,        100156,      258972,      56628;
[8]    26333,       649950,      3288327,     2668750;
[9]    106762,      3944928,     34374186,    66449432,    12317877;
[10]   431910,      22764165,    313530000,   1171704435,  792534015;
[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[Table[Q[n, 2, g], {g, 0, (n+1)/2-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

Prog

(PARI)
N = 10; F = 2; 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)));
concat(vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F))))

Crossrefs

Columns k=0-1 give: A000346, A006295.

Sequence in context: A225846 A247937 A343840 * A209049 A217444 A063619

Adjacent sequences: A270403 A270404 A270405 * A270407 A270408 A270409

Keyword

nonn,tabf

Author

Gheorghe Coserea, Mar 16 2016

Status

approved