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A288081 a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 3.

%I A288081 #15 Oct 17 2018 05:21:38
%S A288081 20465052608,2079913241120,104395235785256,3505018618003600,
%T A288081 89390908732820144,1857975645023518752,32904419378927915376,
%U A288081 511895831411154922176,7151648337964982801760,91230456810047671200128,1076401288635137599528944,11867194568934207062990560
%N A288081 a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 3.
%H A288081 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 A288081 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288081 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 A288081 a[n_] := Q[n, 7, 3];
%t A288081 Table[a[n], {n, 12, 27}] (* _Jean-François Alcover_, Oct 17 2018 *)
%o A288081 (PARI)
%o A288081 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288081 A288081_ser(N) = {
%o A288081   my(y = A000108_ser(N+1));
%o A288081   -8*y*(y-1)^12*(14699198844*y^11 + 323418619692*y^10 + 1093150970776*y^9 - 2010290018547*y^8 - 3822380209098*y^7 + 7160304314725*y^6 - 371305853280*y^5 - 4606441266688*y^4 + 2480182576832*y^3 - 129107145168*y^2 - 150618243904*y + 20945187392)/(y-2)^35;
%o A288081 };
%o A288081 Vec(A288081_ser(12))
%Y A288081 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, this sequence, A288262 f=8, A288263 f=9, A288264 f=10.
%Y A288081 Column 7 of A269923.
%Y A288081 Cf. A000108.
%K A288081 nonn
%O A288081 12,1
%A A288081 _Gheorghe Coserea_, Jun 07 2017