|
%I #14 Oct 16 2018 13:49:31
%S 17302190625720,2089035241981688,123981042854132536,
%T 4892650539994184868,145737674581607574840,3515647035511186627416,
%U 71823371612912533887168,1281537868340178808063824,20423544863369526066131328,295680368360952875467454880,3940377769373862621216994864
%N a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 4.
%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.
%F G.f.: -4*y*(y-1)^14*(18995313191166*y^13 + 602583747147072*y^12 + 3880832501643076*y^11 + 259447266126966*y^10 - 24577880734142257*y^9 + 10075843752456953*y^8 + 45406701745704921*y^7 - 44360505974166179*y^6 - 5860774604042624*y^5 + 22759971294835512*y^4 - 8598423383057104*y^3 - 18688742922288*y^2 + 464831946526080*y - 48608581644864)/(y-2)^41, where y=A000108(x).
%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) ((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}]);
%t a[n_] := Q[n, 7, 4];
%t Table[a[n], {n, 14, 24}] (* _Jean-François Alcover_, Oct 16 2018 *)
%Y 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, A288276 f=6, this sequence, A288278 f=8, A288279 f=9, A288280 f=10.
%Y Column 7 of A269924.
%Y Cf. A000108.
%K nonn
%O 14,1
%A _Gheorghe Coserea_, Jun 08 2017
|
