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A288083 a(n) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus 2.

%I A288083 #16 Oct 18 2018 03:03:46
%S A288083 6468,258972,5554188,85421118,1059255456,11270290416,106853266632,
%T A288083 925572602058,7454157823560,56532447160536,407653880116680,
%U A288083 2815913391715452,18743188498056288,120789163612555200,756589971284883792,4621041111902656770,27595482540212519064,161490751719681569736
%N A288083 a(n) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus 2.
%H A288083 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 A288083 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288083 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 A288083 a[n_] := Q[n, 3, 2];
%t A288083 Table[a[n], {n, 6, 23}] (* _Jean-François Alcover_, Oct 18 2018 *)
%o A288083 (PARI)
%o A288083 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288083 A288083_ser(N) = {
%o A288083   my(y = A000108_ser(N+1));
%o A288083   -6*y*(y-1)^6*(161*y^5 + 4005*y^4 + 4173*y^3 - 10701*y^2 + 2880*y + 560)/(y-2)^17;
%o A288083 };
%o A288083 Vec(A288083_ser(18))
%Y A288083 Rooted maps of genus 2 with n edges and f faces for 1<=f<=10: A006298 f=1, A288082 f=2, this sequence, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
%Y A288083 Column 3 of A269922, column 2 of A270407.
%Y A288083 Cf. A000108.
%K A288083 nonn
%O A288083 6,1
%A A288083 _Gheorghe Coserea_, Jun 05 2017