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A288082 a(n) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus 2.

%I A288082 #15 Oct 18 2018 03:03:58
%S A288082 483,15018,258972,3288327,34374186,313530000,2583699888,19678611645,
%T A288082 140725699686,955708437684,6216591472728,38985279745230,
%U A288082 236923660397172,1401097546161936,8089830217844928,45732525474843801,253705943922820830,1383896652090932364,7434748752218650632,39394773780853063650
%N A288082 a(n) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus 2.
%H A288082 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 A288082 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288082 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 A288082 a[n_] := Q[n, 2, 2];
%t A288082 Table[a[n], {n, 5, 24}] (* _Jean-François Alcover_, Oct 18 2018 *)
%o A288082 (PARI)
%o A288082 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288082 A288082_ser(N) = {
%o A288082   my(y = A000108_ser(N+1));
%o A288082   3*y*(y-1)^5*(7*y^4 + 294*y^3 + 309*y^2 - 547*y + 98)/(y-2)^14;
%o A288082 };
%o A288082 Vec(A288082_ser(20))
%Y A288082 Rooted maps of genus 2 with n edges and f faces for 1<=f<=10: A006298 f=1, this sequence, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
%Y A288082 Column 2 of A269922, column 2 of A270406.
%Y A288082 Cf. A000108.
%K A288082 nonn
%O A288082 5,1
%A A288082 _Gheorghe Coserea_, Jun 05 2017