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A288078 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 3.

%I A288078 #14 Oct 17 2018 05:22:12
%S A288078 17454580,1171704435,40121261136,945068384880,17326957790896,
%T A288078 264477214235234,3505018618003600,41491242915292306,
%U A288078 447708887118504600,4470547991985864322,41790549086980226368,369061676845849000520,3101645444966543203008,24954084939131951164980,193145505023621965434976,1444143475412182351017494,10467259286591304015806600
%N A288078 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 3.
%H A288078 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 A288078 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288078 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 A288078 a[n_] := Q[n, 4, 3];
%t A288078 Table[a[n], {n, 9, 26}] (* _Jean-François Alcover_, Oct 17 2018 *)
%o A288078 (PARI)
%o A288078 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288078 A288078_ser(N) = {
%o A288078   my(y = A000108_ser(N+1));
%o A288078   y*(y-1)^9*(5008230*y^8 + 164100330*y^7 + 620429875*y^6 - 742482075*y^5 - 1203385090*y^4 + 1546511666*y^3 - 224365292*y^2 - 189952744*y + 41589680)/(y-2)^26;
%o A288078 };
%o A288078 Vec(A288078_ser(17))
%Y A288078 Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, this sequence, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
%Y A288078 Column 4 of A269923.
%Y A288078 Cf. A000108.
%K A288078 nonn
%O A288078 9,1
%A A288078 _Gheorghe Coserea_, Jun 07 2017