A288088 - OEIS

%I #15 Oct 18 2018 03:51:32

%S 205633428,19678611645,925572602058,29079129795702,694146691745820,

%T 13518984452463630,224686278407291148,3286157560248860532,

%U 43241609165618454096,520516978029736518606,5805858136761540452700,60619447491266688750132,597358002436877437320936,5593151345725345725640044

%N a(n) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus 2.

%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, 8, 2];

%t Table[a[n], {n, 11, 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 A288088_ser(N) = {

%o   my(y = A000108_ser(N+1));

%o   3*y*(y-1)^11*(1455480376*y^10 + 18151410348*y^9 + 12284790745*y^8 - 111454641175*y^7 + 46880062914*y^6 + 129967691724*y^5 - 125047028168*y^4 + 14650142480*y^3 + 19075464224*y^2 - 6255822912*y + 360993920)/(y-2)^32;

%o };

%o Vec(A288088_ser(14))

%Y 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, this sequence, A288089 f=9, A288090 f=10.

%Y Column 8 of A269922, column 2 of A270412.

%Y Cf. A000108.

%K nonn

%O 11,1

%A _Gheorghe Coserea_, Jun 05 2017