A269925 - OEIS

%I #23 Aug 10 2018 17:36:44

%S 59520825,4304016990,4304016990,158959754226,354949166565,

%T 158959754226,4034735959800,14805457339920,14805457339920,

%U 4034735959800,79553497760100,420797306522502,691650582088536,420797306522502,79553497760100,1302772718028600,9220982517965400,21853758736216200,21853758736216200,9220982517965400,1302772718028600

%N 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 5.

%C Row n contains n-9 terms.

%H Gheorghe Coserea, <a href="/A269925/b269925.txt">Rows n = 10..210, flattened</a>

%H Sean R. Carrell, Guillaume Chapuy, <a href="http://arxiv.org/abs/1402.6300">Simple recurrence formulas to count maps on orientable surfaces</a>, arXiv:1402.6300 [math.CO], 2014.

%e Triangle starts:

%e n\f  [1]             [2]             [3]             [4]

%e [10] 59520825;

%e [11] 4304016990,     4304016990;

%e [12] 15895975422,    354949166565,   158959754226;

%e [13] 4034735959800,  14805457339920, 14805457339920, 4034735959800;

%e [14] ...

%t Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;

%t 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 Table[Q[n, f, 5], {n, 10, 15}, {f, 1, n-9}] // Flatten (* _Jean-François Alcover_, Aug 10 2018 *)

%o (PARI)

%o N = 15; G = 5; gmax(n) = min(n\2, G);

%o Q = matrix(N + 1, N + 1);

%o Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };

%o Qset(n, g, v) = { Q[n+1, g+1] = v };

%o Quadric({x=1}) = {

%o   Qset(0, 0, x);

%o   for (n = 1, length(Q)-1, for (g = 0, gmax(n),

%o     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),

%o        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),

%o        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,

%o        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));

%o     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));

%o };

%o Quadric('x);

%o concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

%Y Rooted maps of genus 5 with n edges and f faces for 1<=f<=10: A288281 f=1, A288282 f=2, A288283 f=3, A288284 f=4, A288285 f=5, A288286 f=6, A288287 f=7, A288288 f=8, A288289 f=9, A288290 f=10.

%Y Row sums give A238355 (column 5 of A269919).

%Y Cf. A035309, A269921, A269922, A269923, A269924, A270406, A270407, A270408, A270409, A270410, A270411, A270412.

%K nonn,tabl

%O 10,1

%A _Gheorghe Coserea_, Mar 15 2016