A270407 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus g.

2, 22, 164, 70, 1030, 1720, 5868, 24164, 6468, 31388, 256116, 258972, 160648, 2278660, 5554188, 1169740, 795846, 17970784, 85421118, 66449432, 3845020, 129726760, 1059255456, 1955808460, 351683046, 18211380, 875029804, 11270290416, 40121261136, 26225260226

Offset

2,1

Comments

Row n contains floor(n/2) terms.

Links

Gheorghe Coserea, Rows n = 2..102, 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]
[2]    2;
[3]    22;
[4]    164,         70;
[5]    1030,        1720;
[6]    5868,        24164,       6468;
[7]    31388,       256116,      258972;
[8]    160648,      2278660,     5554188,     1169740;
[9]    795846,      17970784,    85421118,    66449432;
[10]   3845020,     129726760,   1059255456,  1955808460,  351683046;
[11]   18211380,    875029804,   11270290416, 40121261136, 26225260226;
[12]   ...

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, 3, g];
Table[T[n, g], {n, 2, 11}, {g, 0, Quotient[n, 2]-1}] // Flatten (* Jean-François Alcover, Oct 18 2018 *)

Prog

(PARI)
N = 11; F = 3; 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: A000184, A006296.

Sequence in context: A091169 A279380 A230835 * A000184 A007613 A346796

Adjacent sequences: A270404 A270405 A270406 * A270408 A270409 A270410

Keyword

nonn,tabf

Author

Gheorghe Coserea, Mar 16 2016

Status

approved