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%I #16 Oct 18 2018 03:02:53
%S 420,14065,256116,3392843,36703824,344468530,2908358552,22620890127,
%T 164767964504,1137369687454,7506901051000,47700234551918,
%U 293370096957504,1753945289216484,10229201477344752,58364244137596695,326571194881454376,1795631576981016038,9718877491130813368,51858415558095569962
%N a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 1.
%H 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 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, 4, 1];
%t Table[a[n], {n, 5, 24}] (* _Jean-François Alcover_, Oct 18 2018 *)
%o (PARI)
%o A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288071_ser(N) = {
%o my(y = A000108_ser(N+1));
%o y*(y-1)^5*(307*y^4 + 2411*y^3 - 2094*y^2 - 708*y + 504)/(y-2)^14;
%o };
%o Vec(A288071_ser(20))
%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, this sequence, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
%Y Column 4 of A269921.
%Y Cf. A000108.
%K nonn
%O 5,1
%A _Gheorghe Coserea_, Jun 05 2017
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