A270412 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus g.

429, 26333, 795846, 291720, 16322085, 22764165, 259477218, 875029804, 205633428, 3435601554, 22620890127, 19678611645, 39599553708, 448035881592, 925572602058, 174437377400, 409230997461, 7302676928666, 29079129795702, 19925913354061

Offset

7,1

Comments

Row n contains floor((n-5)/2) terms.

Links

Gheorghe Coserea, Rows n = 7..107, 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]
[7]    429;
[8]    26333;
[9]    795846,          291720;
[10]   16322085,        22764165;
[11]   259477218,       875029804,       205633428;
[12]   3435601554,      22620890127,     19678611645;
[13]   39599553708,     448035881592,    925572602058,    174437377400;
[14]   409230997461,    7302676928666,   29079129795702,  19925913354061;
[15]   ...

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, 8, g];
Table[T[n, g], {n, 7, 14}, {g, 0, Quotient[n-5, 2]-1}] // Flatten (* Jean-François Alcover, Oct 18 2018 *)

Prog

(PARI)
N = 14; F = 8; 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)));
v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));
concat(v)

Crossrefs

Cf. A270411.

Sequence in context: A145056 A064304 A264180 * A258494 A258395 A215547

Adjacent sequences: A270409 A270410 A270411 * A270413 A270414 A270415

Keyword

nonn,tabf

Author

Gheorghe Coserea, Mar 17 2016

Status

approved