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%I #14 Oct 16 2018 13:58:14
%S 182231849209410,24325590127655531,1587135819804394530,
%T 68503375296263488977,2221381417843144801098,58089920897558352891672,
%U 1281537868340178808063824,24605894500188479477960928,420612140517667008915254376,6512251870890866709301451550,92559480623350598649493386580
%N a(n) is the number of rooted maps with n edges and 8 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.: y*(y-1)^15*(2141204115631518*y^14 + 62802256981978404*y^13 + 390904315702808387*y^12 - 17469926941849537*y^11 - 2715522908192830943*y^10 + 1209526054185992549*y^9 + 5862111891800632315*y^8 - 6084780630540788053*y^7 - 1344178041537337418*y^6 + 4359417524034703460*y^5 - 1779344954166712472*y^4 - 128701285301543888*y^3 + 220665627694548576*y^2 - 38233669153240512*y + 844773167217024)/(y-2)^44, 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, 8, 4];
%t Table[a[n], {n, 15, 25}] (* _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, this sequence, A288279 f=9, A288280 f=10.
%Y Column 8 of A269924.
%Y Cf. A000108.
%K nonn
%O 15,1
%A _Gheorghe Coserea_, Jun 08 2017
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