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A288077 a(n) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus 3.

%I A288077 #14 Oct 17 2018 05:22:31
%S A288077 1169740,66449432,1955808460,40121261136,647739636160,8789123742880,
%T A288077 104395235785256,1115525500250760,10933959720960760,99727841192820016,
%U A288077 855779329367736840,6968569097113244096,54217755730994858080,405300088876353160320,2924455840981270327952,20446207814548586119000,138958722742591452843432
%N A288077 a(n) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus 3.
%H A288077 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.
%t A288077 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288077 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 A288077 a[n_] := Q[n, 3, 3];
%t A288077 Table[a[n], {n, 8, 25}] (* _Jean-François Alcover_, Oct 17 2018 *)
%o A288077 (PARI)
%o A288077 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288077 A288077_ser(N) = {
%o A288077   my(y = A000108_ser(N+1));
%o A288077   -4*y*(y-1)^8*(28314*y^7 + 1229985*y^6 + 4821650*y^5 - 4914053*y^4 - 6967314*y^3 + 7429165*y^2 - 1071576*y - 263736)/(y-2)^23;
%o A288077 };
%o A288077 Vec(A288077_ser(17))
%Y A288077 Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, this sequence, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
%Y A288077 Column 3 of A269923.
%Y A288077 Cf. A000108.
%K A288077 nonn
%O A288077 8,1
%A A288077 _Gheorghe Coserea_, Jun 07 2017