OFFSET
0,2
COMMENTS
Row n contains floor((n+2)/2) terms.
Equivalently, T(n,g) is the number of rooted bipartite quadrangulations with n faces of an orientable surface of genus g.
LINKS
Gheorghe Coserea, Rows n = 0..200, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.
FORMULA
(n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.
EXAMPLE
Triangle starts:
n\g [0] [1] [2] [3] [4]
[0] 1;
[1] 2;
[2] 9, 1;
[3] 54, 20;
[4] 378, 307, 21;
[5] 2916, 4280, 966;
[6] 24057, 56914, 27954, 1485;
[7] 208494, 736568, 650076, 113256;
[8] 1876446, 9370183, 13271982, 5008230, 225225;
[9] 17399772, 117822512, 248371380, 167808024, 24635754;
[10] ...
MATHEMATICA
T[0, 0] = 1; T[n_, g_] /; g<0 || g>n/2 = 0; T[n_, g_] := T[n, g] = ((4n-2)/ 3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n-k)- 1) T[k-1, i] T[n-k-1, g-i], {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n/2}] // Flatten (* Jean-François Alcover, Jul 20 2018 *)
PROG
(PARI)
N = 9; 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, N, 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();
concat(vector(N+1, n, vector(1 + gmax(n-1), g, Qget(n-1, g-1))))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, Mar 07 2016
STATUS
approved