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%I #14 Oct 17 2018 05:22:46
%S 56628,2668750,66449432,1171704435,16476937840,196924458720,
%T 2079913241120,19925913354061,176357530955320,1461629029629340,
%U 11460411934448048,85694099173907510,614960028331370816,4257157940494918160,28549761695867223680,186131532080726321441,1183191417356212860200,7351865732351585503652
%N a(n) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus 3.
%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) ((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 a[n_] := Q[n, 2, 3];
%t Table[a[n], {n, 7, 24}] (* _Jean-François Alcover_, Oct 17 2018 *)
%o (PARI)
%o A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288076_ser(N) = {
%o my(y = A000108_ser(N+1));
%o y*(y-1)^7*(1485*y^6 + 111969*y^5 + 453295*y^4 - 389693*y^3 - 443894*y^2 + 361702*y - 38236)/(y-2)^20;
%o };
%o Vec(A288076_ser(18))
%Y Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, this sequence, 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 Column 2 of A269923.
%Y Cf. A000108.
%K nonn
%O 7,1
%A _Gheorghe Coserea_, Jun 07 2017
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