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A288086 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 2.

%I A288086 #15 Oct 18 2018 03:34:39
%S A288086 4390386,313530000,11270290416,276221817810,5235847653036,
%T A288086 82234427131416,1117259292848016,13518984452463630,148755268498286436,
%U A288086 1511718920778951024,14358354462488121408,128656798319026864068,1095747149735034238680,8924653047010981590288,69866689045523025725664
%N A288086 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 2.
%H A288086 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 A288086 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288086 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 A288086 a[n_] := Q[n, 6, 2];
%t A288086 Table[a[n], {n, 9, 23}] (* _Jean-François Alcover_, Oct 18 2018 *)
%o A288086 (PARI)
%o A288086 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288086 A288086_ser(N) = {
%o A288086   my(y = A000108_ser(N+1));
%o A288086   6*y*(y-1)^9*(2211997*y^8 + 32071458*y^7 + 27414609*y^6 - 154896511*y^5 + 58087530*y^4 + 94331624*y^3 - 68497296*y^2 + 8775424*y + 1232896)/(y-2)^26;
%o A288086 };
%o A288086 Vec(A288086_ser(15))
%Y A288086 Rooted maps of genus 2 with n edges and f faces for 1<=f<=10: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, this sequence, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
%Y A288086 Column 6 of A269922, column 2 of A270410.
%Y A288086 Cf. A000108.
%K A288086 nonn
%O A288086 9,1
%A A288086 _Gheorghe Coserea_, Jun 05 2017