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%I #32 Aug 10 2018 17:36:22
%S 21,483,483,6468,15018,6468,66066,258972,258972,66066,570570,3288327,
%T 5554188,3288327,570570,4390386,34374186,85421118,85421118,34374186,
%U 4390386,31039008,313530000,1059255456,1558792200,1059255456,313530000,31039008
%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 2.
%C Row n contains n-3 terms.
%H Gheorghe Coserea, <a href="/A269922/b269922.txt">Rows n = 4..204, 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]
%e [4] 21;
%e [5] 483, 483;
%e [6] 6468, 15018, 6468;
%e [7] 66066, 258972, 258972, 66066;
%e [8] 570570, 3288327, 5554188, 3288327, 570570;
%e [9] 4390386, 34374186, 85421118, 85421118, 34374186, 4390386;
%e [10] ...
%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, 2], {n, 4, 10}, {f, 1, n-3}] // Flatten (* _Jean-François Alcover_, Aug 10 2018 *)
%o (PARI)
%o N = 10; G = 2; 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 f=1-10 give: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
%Y Row sums give A006301 (column 2 of A269919).
%Y Cf. A006299 (row maxima), A269921.
%K nonn,tabl
%O 4,1
%A _Gheorghe Coserea_, Mar 15 2016
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