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%I #15 Oct 17 2018 11:22:41
%S 174437377400,19925913354061,1115525500250760,41491242915292306,
%T 1165172136542282424,26522236056202555206,511895831411154922176,
%U 8640883781524178188980,130468023103972196647776,1792206112041706943912462,22695416350294243544684240,267740228837597817351215676
%N a(n) is the number of rooted maps with n edges and 8 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) ((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, 8, 3];
%t Table[a[n], {n, 13, 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 A288262_ser(N) = {
%o my(y = A000108_ser(N+1));
%o y*(y-1)^13*(2675326679856*y^12 + 54684388381464*y^11 + 178122315841075*y^10 - 372236561648447*y^9 - 717438005317146*y^8 + 1482970059363466*y^7 - 17264319660476*y^6 - 1294789702753096*y^5 + 770104389507952*y^4 - 4493523304288*y^3 - 105563098094272*y^2 + 24298454684800*y - 895286303488)/(y-2)^38;
%o };
%o Vec(A288262_ser(12))
%Y Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, this sequence, A288263 f=9, A288264 f=10.
%Y Column 8 of A269923.
%Y Cf. A000108.
%K nonn
%O 13,1
%A _Gheorghe Coserea_, Jun 07 2017
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