OFFSET
5,1
COMMENTS
Row n contains floor((n-3)/2) terms.
LINKS
Gheorghe Coserea, Rows n = 5..105, 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]
[5] 42;
[6] 1586;
[7] 31388, 12012;
[8] 442610, 649950;
[9] 5030004, 17970784, 4390386;
[10] 49145460, 344468530, 313530000;
[11] 429166584, 5188948072, 11270290416, 2198596400;
[12] 3435601554, 65723863196, 276221817810, 196924458720;
[13] ...
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, 6, g];
Table[T[n, g], {n, 5, 13}, {g, 0, Quotient[n-3, 2]-1}] // Flatten (* Jean-François Alcover, Oct 18 2018 *)
PROG
(PARI)
N = 12; F = 6; 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
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, Mar 17 2016
STATUS
approved