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%I #15 Oct 16 2018 13:58:05
%S 15894791312284170,2548272396065512974,197822824662547694148,
%T 10071757699155275906824,382217975972687580876304,
%U 11612741439751867739074432,295680368360952875467454880,6512251870890866709301451550,126977551039680427095997314540,2230836871835420574103711453068
%N a(n) is the number of rooted maps with n edges and 10 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.: 2*y*(y-1)^17*(667113335854505289*y^16 + 17412039201241985652*y^15 + 101949739105950626070*y^14 - 30202970169901595562*y^13 - 833532476362240891879*y^12 + 447114036864981439647*y^11 + 2316066844919602997013*y^10 - 2673632819222127570107*y^9 - 1088786810085394834566*y^8 + 3157924186313124711792*y^7 - 1371258409341666011952*y^6 - 433458368694714259536*y^5 + 515333809963509426144*y^4 - 126279314363368987008*y^3 - 3637814234318456832*y^2 + 4694513255143047936*y - 365353090019990016)/(y-2)^50, 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) (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] (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, 10, 4];
%t Table[a[n], {n, 17, 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, A288279 f=9, this sequence.
%Y Column 10 of A269924.
%Y Cf. A000108.
%K nonn
%O 17,1
%A _Gheorghe Coserea_, Jun 08 2017
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