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%I #19 Oct 17 2018 21:45:55
%S 12012,649950,17970784,344468530,5188948072,65723863196,729734918432,
%T 7302676928666,67173739068760,576218752277476,4660202610532480,
%U 35839052357422132,263868150558327376,1870153808268516280,12816868756802256832,85256107136168684650,552171259884681058744
%N a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 1.
%H Sean R. Carrell and 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.
%H Sean R. Carrell and Guillaume Chapuy, <a href="https://doi.org/10.1016/j.jcta.2015.01.005">Simple recurrence formulas to count maps on orientable surfaces</a>, Journal of Combinatorial Theory, Series A, 133 (2015), 58-75.
%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, 6, 1];
%t Table[a[n], {n, 7, 23}] (* _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 A287046_ser(N) = {
%o my(y = A000108_ser(N+1));
%o 2*y*(y-1)^7*(28457*y^6 + 179171*y^5 - 222214*y^4 - 172512*y^3 + 257232*y^2 - 59904*y - 4224)/(y-2)^20;
%o };
%o Vec(A287046_ser(17))
%Y Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, this sequence, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
%Y Column 6 of A269921, column 1 of A270410.
%Y Cf. A000108.
%K nonn
%O 7,1
%A _Gheorghe Coserea_, Jun 04 2017
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