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%I #15 Oct 18 2018 03:51:41
%S 31039008,2583699888,106853266632,2979641557620,63648856688592,
%T 1117259292848016,16842445235560944,224686278407291148,
%U 2710382626755160416,30044423965980553536,309859885439753598768,3002524783711124880936,27551689577648333614176,240961534103705377359840,2019318707410947848445792
%N a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 2.
%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.
%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) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 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] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
%t a[n_] := Q[n, 7, 2];
%t Table[a[n], {n, 10, 24}] (* _Jean-François Alcover_, Oct 18 2018 *)
%o (PARI)
%o A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288087_ser(N) = {
%o my(y = A000108_ser(N+1));
%o -12*y*(y-1)^10*(20697615*y^9 + 275716321*y^8 + 211910021*y^7 - 1514443109*y^6 + 601694224*y^5 + 1328709592*y^4 - 1136750032*y^3 + 153705072*y^2 + 76788992*y - 15442112)/(y-2)^29;
%o };
%o Vec(A288087_ser(15))
%Y Rooted maps of genus 2 with n edges and f faces for 1<=f<=10: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, this sequence, A288088 f=8, A288089 f=9, A288090 f=10.
%Y Column 7 of A269922, column 2 of A270411.
%Y Cf. A000108.
%K nonn
%O 10,1
%A _Gheorghe Coserea_, Jun 05 2017
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