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A288080 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 3.

%I A288080 #15 Oct 17 2018 05:21:48
%S A288080 2198596400,196924458720,8789123742880,264477214235234,
%T A288080 6087558311398000,114899070275212424,1857975645023518752,
%U A288080 26522236056202555206,341505418008822731328,4031165546220945277040,44171448380277095027584,453764845712090669861060,4405234525240663358548000,40682085269643556632419504,359336179016097679450360000
%N A288080 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 3.
%H A288080 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 A288080 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288080 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 A288080 a[n_] := Q[n, 6, 3];
%t A288080 Table[a[n], {n, 11, 28}] (* _Jean-François Alcover_, Oct 17 2018 *)
%o A288080 (PARI)
%o A288080 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288080 A288080_ser(N) = {
%o A288080   my(y = A000108_ser(N+1));
%o A288080   2*y*(y-1)^11*(2360692395*y^10 + 57065162931*y^9 + 200199438395*y^8 - 321653197109*y^7 - 594662939878*y^6 + 999754510326*y^5 - 90653073868*y^4 - 435707439920*y^3 + 201952082336*y^2 - 14180151168*y - 3375786240)/(y-2)^32;
%o A288080 };
%o A288080 Vec(A288080_ser(15))
%Y A288080 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, this sequence, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
%Y A288080 Column 6 of A269923.
%Y A288080 Cf. A000108.
%K A288080 nonn
%O A288080 11,1
%A A288080 _Gheorghe Coserea_, Jun 07 2017