7,1

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

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.

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] ...

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 *)

(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)

Cf. A270411.

Sequence in context: A064304 A264180 A393925 * A258494 A258395 A215547

Adjacent sequences: A270409 A270410 A270411 * A270413 A270414 A270415

nonn,tabf

Gheorghe Coserea, Mar 17 2016

approved