A288073 - OEIS

%I #17 Nov 05 2025 15:22:39

%S 1385670,126264820,5593305476,164767964504,3682811916980,

%T 67173739068760,1046677747672360,14373136466094880,177882700353757460,

%U 2017523504473479992,21241931655650633720,209732362862241103248,1957830216739337392584,17394726697224718134384,147908195064869691109072

%N a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 1.

%H Sean R. Carrell, Guillaume Chapuy, <a href="https://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, 9, 1];

%t Table[a[n], {n, 10, 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 A288073_ser(N) = {

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

%o   -2*y*(y-1)^10*(58911256*y^9 + 315266323*y^8 - 563073084*y^7 - 706445836*y^6 + 1588166368*y^5 - 488205920*y^4 - 472512192*y^3 + 315108288*y^2 - 44342784*y - 2179584)/(y-2)^29;

%o };

%o Vec(A288073_ser(17))

%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, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, this sequence, A288074 f=10.

%Y Column 9 of A269921.

%K nonn

%O 10,1

%A _Gheorghe Coserea_, Jun 05 2017