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%I #14 Oct 16 2018 13:48:37
%S 1763184571730010,258634264294653390,18451302662846918700,
%T 866831237081712285138,30468100266480917147760,
%U 860337164444236894357488,20423544863369526066131328,420612140517667008915254376,7689357064107454375292572788,126977551039680427095997314540,1920060399356995304343259728312
%N a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 4.
%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.
%F G.f.: -6*y*(y-1)^16*(9225445001552610*y^15 + 253889174613116085*y^14 + 1531144661703557241*y^13 - 254390452688914375*y^12 - 11576322921612113581*y^11 + 5646113444605154169*y^10 + 28587502564009313669*y^9 - 31350769849259642447*y^8 - 9832935993984430480*y^7 + 29500732589692418132*y^6 - 12567984363713561312*y^5 - 2218978200544343392*y^4 + 2888444088307833216*y^3 - 630076702195212352*y^2 + 8436883230156800*y + 6263496930404352)/(y-2)^47, where y=A000108(x).
%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) ((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 a[n_] := Q[n, 9, 4];
%t Table[a[n], {n, 16, 26}] (* _Jean-François Alcover_, Oct 16 2018 *)
%Y 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, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, this sequence, A288280 f=10.
%Y Column 9 of A269924.
%Y Cf. A000108.
%K nonn
%O 16,1
%A _Gheorghe Coserea_, Jun 08 2017
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