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A288089 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 2.

%I A288089 #14 Oct 18 2018 03:51:23
%S A288089 1293938646,140725699686,7454157823560,261637840342860,
%T A288089 6928413234959820,148755268498286436,2710382626755160416,
%U A288089 43241609165618454096,617910462111714896820,8044640800289827566756,96690983139765469347024,1084226645505246141589704,11439196912435362172792536,114351801899024314438876200
%N A288089 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 2.
%H A288089 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 A288089 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288089 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 A288089 a[n_] := Q[n, 9, 2];
%t A288089 Table[a[n], {n, 12, 25}] (* _Jean-François Alcover_, Oct 18 2018 *)
%o A288089 (PARI)
%o A288089 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288089 A288089_ser(N) = {
%o A288089   my(y = A000108_ser(N+1));
%o A288089   -6*y*(y-1)^12*(12205186004*y^11 + 144345246789*y^10 + 83883548039*y^9 - 978172313331*y^8 + 436600889944*y^7 + 1435650005364*y^6 - 1511798886368*y^5 + 121539026592*y^4 + 411304907520*y^3 - 171035694144*y^2 + 14120686592*y + 1573053440)/(y-2)^35;
%o A288089 };
%o A288089 Vec(A288089_ser(13))
%Y A288089 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, A288086 f=6, A288087 f=7, A288088 f=8, this sequence, A288090 f=10.
%Y A288089 Column 9 of A269922.
%Y A288089 Cf. A000108.
%K A288089 nonn
%O A288089 12,1
%A A288089 _Gheorghe Coserea_, Jun 05 2017