A270408 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus g.

5, 93, 1030, 420, 8885, 14065, 65954, 256116, 66066, 442610, 3392843, 3288327, 2762412, 36703824, 85421118, 17454580, 16322085, 344468530, 1558792200, 1171704435, 92400330, 2908358552, 22555934280, 40121261136, 7034538511, 505403910, 22620890127, 276221817810, 945068384880, 600398249550

Offset

3,1

Comments

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

Links

Gheorghe Coserea, Rows n = 3..103, 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]
[3]    5;
[4]    93;
[5]    1030,        420;
[6]    8885,        14065;
[7]    65954,       256116,      66066;
[8]    442610,      3392843,     3288327;
[9]    2762412,     36703824,    85421118,    17454580;
[10]   16322085,    344468530,   1558792200,  1171704435;
[11]   92400330,    2908358552,  22555934280, 40121261136, 7034538511;
[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, 4, g];
Table[T[n, g], {n, 3, 12}, {g, 0, Quotient[n-1, 2]-1}] // Flatten (* Jean-François Alcover, Oct 18 2018 *)

Prog

(PARI)
N = 11; F = 4; 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

Cf. A000365 (column 0).

Sequence in context: A295407 A152283 A205344 * A000365 A209471 A012784

Adjacent sequences: A270405 A270406 A270407 * A270409 A270410 A270411

Keyword

nonn,tabf

Author

Gheorghe Coserea, Mar 17 2016

Status

approved