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A269923
Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 3.
13
1485, 56628, 56628, 1169740, 2668750, 1169740, 17454580, 66449432, 66449432, 17454580, 211083730, 1171704435, 1955808460, 1171704435, 211083730, 2198596400, 16476937840, 40121261136, 40121261136, 16476937840, 2198596400, 20465052608, 196924458720, 647739636160
OFFSET
6,1
COMMENTS
Row n contains n-5 terms.
LINKS
Gheorghe Coserea, Rows n = 6..206, 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\f [1] [2] [3] [4] [5]
[6] 1485;
[7] 56628, 56628;
[8] 1169740, 2668750, 1169740;
[9] 17454580, 66449432, 66449432, 17454580;
[10] 211083730, 1171704435, 1955808460, 1171704435, 211083730;
[11] ...
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}]);
Table[Q[n, f, 3], {n, 6, 12}, {f, 1, n-5}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
PROG
(PARI)
N = 12; G = 3; gmax(n) = min(n\2, G);
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);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))
CROSSREFS
Columns f=1-10 give: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
Row sums give A104742 (column 3 of A269919).
Sequence in context: A045008 A327880 A257713 * A288075 A104742 A278853
KEYWORD
nonn,tabl
AUTHOR
Gheorghe Coserea, Mar 15 2016
STATUS
approved