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A288275 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 4.

%I A288275 #14 Oct 16 2018 13:49:20
%S A288275 111159740692,10743797911132,517592962672296,16789118602155860,
%T A288275 415691294404230748,8419549939292302908,145737674581607574840,
%U A288275 2221381417843144801098,30468100266480917147760,382217975972687580876304,4441222132558609054169216,48280421251792089554320464
%N A288275 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 4.
%H A288275 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.
%F A288275 G.f.: -2*y*(y-1)^12*(33259798737*y^11 + 1329990099093*y^10 + 9262655718313*y^9 + 2336641955449*y^8 - 47227883527259*y^7 + 17056753299711*y^6 + 58186472373731*y^5 - 48817840576153*y^4 + 819511081872*y^3 + 9462230411332*y^2 - 2475017890416*y + 88807125936)/(y-2)^35, where y=A000108(x).
%t A288275 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288275 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 A288275 a[n_] := Q[n, 5, 4];
%t A288275 Table[a[n], {n, 12, 23}] (* _Jean-François Alcover_, Oct 16 2018 *)
%Y A288275 Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, this sequence, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
%Y A288275 Column 5 of A269924.
%Y A288275 Cf. A000108.
%K A288275 nonn
%O A288275 12,1
%A A288275 _Gheorghe Coserea_, Jun 08 2017