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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.

%I A269923 #26 Aug 10 2018 17:36:29
%S A269923 1485,56628,56628,1169740,2668750,1169740,17454580,66449432,66449432,
%T A269923 17454580,211083730,1171704435,1955808460,1171704435,211083730,
%U A269923 2198596400,16476937840,40121261136,40121261136,16476937840,2198596400,20465052608,196924458720,647739636160
%N 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.
%C A269923 Row n contains n-5 terms.
%H A269923 Gheorghe Coserea, <a href="/A269923/b269923.txt">Rows n = 6..206, flattened</a>
%H A269923 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 A269923 Triangle starts:
%e A269923 n\f  [1]          [2]          [3]          [4]          [5]
%e A269923 [6]  1485;
%e A269923 [7]  56628,       56628;
%e A269923 [8]  1169740,     2668750,     1169740;
%e A269923 [9]  17454580,    66449432,    66449432,    17454580;
%e A269923 [10] 211083730,   1171704435,  1955808460,  1171704435,  211083730;
%e A269923 [11] ...
%t A269923 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;
%t A269923 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 A269923 Table[Q[n, f, 3], {n, 6, 12}, {f, 1, n-5}] // Flatten (* _Jean-François Alcover_, Aug 10 2018 *)
%o A269923 (PARI)
%o A269923 N = 12; G = 3; gmax(n) = min(n\2, G);
%o A269923 Q = matrix(N + 1, N + 1);
%o A269923 Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
%o A269923 Qset(n, g, v) = { Q[n+1, g+1] = v };
%o A269923 Quadric({x=1}) = {
%o A269923   Qset(0, 0, x);
%o A269923   for (n = 1, length(Q)-1, for (g = 0, gmax(n),
%o A269923     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
%o A269923        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
%o A269923        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
%o A269923        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
%o A269923     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
%o A269923 };
%o A269923 Quadric('x);
%o A269923 concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))
%Y A269923 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.
%Y A269923 Row sums give A104742 (column 3 of A269919).
%Y A269923 Cf. A269921, A269922, A269924, A269925.
%K A269923 nonn,tabl
%O A269923 6,1
%A A269923 _Gheorghe Coserea_, Mar 15 2016