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%I #13 Oct 17 2018 12:33:02
%S 79553497760100,9220982517965400,528887751025584600,
%T 20269771718252599536,588564117958709029644,13881153040572190501512,
%U 277921666244135490925320,4869474711666664850333856,76330117260895762678976496,1088463806617771584122226336,14304840156674599302991391808,175067544404400195382759080000
%N a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 5.
%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.: -12*y*(y-1)^14*(3140032216620*y^13 + 168745438117215*y^12 + 1823095410398560*y^11 + 3655757687054272*y^10 - 10735527168335100*y^9 - 13611993085165141*y^8 + 33238393245141476*y^7 - 1171322344070974*y^6 - 27716201280764020*y^5 + 15575605858027959*y^4 + 683444198956148*y^3 - 2374578542797076*y^2 + 479239083620192*y - 11169074253456)/(y-2)^41, 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) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 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] (2 k - 1) (2 l - 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, 5, 5];
%t Table[a[n], {n, 14, 25}] (* _Jean-François Alcover_, Oct 17 2018 *)
%Y Rooted maps of genus 5 with n edges and f faces for 1<=f<=10: A288281 f=1, A288282 f=2, A288283 f=3, A288284 f=4, this sequence, A288286 f=6, A288287 f=7, A288288 f=8, A288289 f=9, A288290 f=10.
%Y Column 5 of A269925.
%Y Cf. A000108.
%K nonn
%O 14,1
%A _Gheorghe Coserea_, Jun 11 2017
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