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MATHEMATICA
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Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1)((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n-1, f-1, g] + (2n-3)(2n-2)(2n-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] (2k-1)(2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 6, 4];
Table[a[n], {n, 13, 24}] (* Jean-François Alcover, Oct 16 2018 *)
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CROSSREFS
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Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, this sequence, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Column 6 of A269924.
Cf. A000108.
Sequence in context: A335085 A213601 A213646 * A246251 A320943 A248892
Adjacent sequences: A288273 A288274 A288275 * A288277 A288278 A288279
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