%I #40 Aug 10 2018 17:36:15
%S 1,1,1,2,5,2,5,22,22,5,14,93,164,93,14,42,386,1030,1030,386,42,132,
%T 1586,5868,8885,5868,1586,132,429,6476,31388,65954,65954,31388,6476,
%U 429,1430,26333,160648,442610,614404,442610,160648,26333,1430
%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 0.
%C Row n contains n+1 terms.
%H Gheorghe Coserea, <a href="/A269920/b269920.txt">Rows n = 0..200, 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] [5] [6] [7] [8]
%e [0] 1;
%e [1] 1, 1;
%e [2] 2, 5, 2;
%e [3] 5, 22, 22, 5;
%e [4] 14, 93, 164, 93, 14;
%e [5] 42, 386, 1030, 1030, 386, 42;
%e [6] 132, 1586, 5868, 8885, 5868, 1586, 132;
%e [7] 429, 6476, 31388, 65954, 65954, 31388, 6476, 429;
%e [8] ...
%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, 0], {n, 0, 8}, {f, 1, n+1}] // Flatten (* _Jean-François Alcover_, Aug 10 2018 *)
%o (PARI)
%o N = 8; G = 0; 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 Columns k=1-6 give: A000108, A000346, A000184, A000365, A000473, A000502.
%Y Row sums give A000168 (column 0 of A269919).
%Y Cf. A006294 (row maxima).
%K nonn,tabl
%O 0,4
%A _Gheorghe Coserea_, Mar 14 2016