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A288090 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 2.

%I A288090 #14 Oct 18 2018 03:51:16
%S A288090 7808250450,955708437684,56532447160536,2200626948631386,
%T A288090 64232028100704156,1511718920778951024,30044423965980553536,
%U A288090 520516978029736518606,8044640800289827566756,112860842135424498808968,1456882832375987896763184,17491588653334242501297012,197038603477850885815215480
%N A288090 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 2.
%H A288090 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 A288090 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A288090 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 A288090 a[n_] := Q[n, 10, 2];
%t A288090 Table[a[n], {n, 13, 25}] (* _Jean-François Alcover_, Oct 18 2018 *)
%o A288090 (PARI)
%o A288090 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A288090 A288090_ser(N) = {
%o A288090   my(y = A000108_ser(N+1));
%o A288090   6*y*(y-1)^13*(197300616213*y^12 + 2233379349250*y^11 + 1077980722075*y^10 - 16537713992125*y^9 + 7856375825902*y^8 + 29387232350368*y^7 - 33290642716432*y^6 + 994024496848*y^5 + 14078465181600*y^4 - 6737013421440*y^3 + 532103069696*y^2 + 244607984896*y - 34798091776)/(y-2)^38;
%o A288090 };
%o A288090 Vec(A288090_ser(13))
%Y A288090 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, A288089 f=9, this sequence.
%Y A288090 Column 10 of A269922.
%Y A288090 Cf. A000108.
%K A288090 nonn
%O A288090 13,1
%A A288090 _Gheorghe Coserea_, Jun 05 2017