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%I #21 Oct 17 2018 05:21:56
%S 211083730,16476937840,647739636160,17326957790896,357391270819604,
%T 6087558311398000,89390908732820144,1165172136542282424,
%U 13767319160210071404,149789855223187292608,1518921342035154605600,14492634832409091816640,131114130730951689447016,1131791523345860091265696,9370402052804684247760928
%N a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 3.
%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.
%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, 5, 3];
%t Table[a[n], {n, 10, 27}] (* _Jean-François Alcover_, Oct 17 2018 *)
%o (PARI)
%o A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288079_ser(N) = {
%o my(y = A000108_ser(N+1));
%o -2*y*(y-1)^10*(83904012*y^9 + 2299548501*y^8 + 8375416306*y^7 - 11663434748*y^6 - 20521873396*y^5 + 30517603222*y^4 - 3781427784*y^3 - 7908127656*y^2 + 2862038656*y - 158105248)/(y-2)^29;
%o };
%o Vec(A288079_ser(15))
%Y Rooted maps of genus 3 with n edges and f faces for 1 <= f <= 10: A288075 (f = 1), A288076 (f = 2), A288077 (f = 3), A288078 (f = 4), this sequence (f = 5), A288080 (f = 6), A288081 (f = 7), A288262 (f = 8), A288263 (f = 9), A288264 (f = 10).
%Y Column 5 of A269923.
%Y Cf. A000108.
%K nonn
%O 10,1
%A _Gheorghe Coserea_, Jun 07 2017
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