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A283812 Poly-Bernoulli numbers B_n^(k) with k = -6.

%I A283812 #31 May 18 2024 14:54:17
%S A283812 1,64,1394,20266,237686,2441314,22934774,202229266,1701740006,
%T A283812 13821281314,109214866454,844558486066,6419351203526,48118995192514,
%U A283812 356641942834934,2618939805811666,19085432672558246,138206899494338914,995563711729120214,7139963278111582066,51017526215427244166
%N A283812 Poly-Bernoulli numbers B_n^(k) with k = -6.
%C A283812 a(n) is also the number of acyclic orientations of the complete bipartite graph K_{6,n}. - _Vincent Pilaud_, Sep 16 2020
%H A283812 Seiichi Manyama, <a href="/A283812/b283812.txt">Table of n, a(n) for n = 0..1179</a>
%H A283812 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (27,-295,1665,-5104,8028,-5040).
%F A283812 a(n) = 720*7^n - 1800*6^n + 1560*5^n - 540*4^n + 62*3^n - 2^n.
%F A283812 From _Colin Barker_, Oct 14 2020: (Start)
%F A283812 G.f.: (1 - x)^2*(1 + 39*x + 38*x^2 - 120*x^3) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
%F A283812 a(n) = 27*a(n-1) - 295*a(n-2) + 1665*a(n-3) - 5104*a(n-4) + 8028*a(n-5) - 5040*a(n-6) for n>5. (End)
%F A283812 E.g.f.: exp(2*x)*(720*exp(5*x) - 1800*exp(4*x) + 1560*exp(3*x) - 540*exp(2*x) + 62*exp(x) - 1). - _Stefano Spezia_, May 18 2024
%t A283812 Table[720*7^n - 1800*6^n + 1560*5^n - 540*4^n + 62*3^n - 2^n , {n, 0, 18}] (* _Indranil Ghosh_, Mar 17 2017 *)
%o A283812 (PARI) a(n) = 720*7^n - 1800*6^n + 1560*5^n - 540*4^n + 62*3^n - 2^n; \\ _Indranil Ghosh_, Mar 17 2017
%o A283812 (PARI) Vec((1 - x)^2*(1 + 39*x + 38*x^2 - 120*x^3) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^20)) \\ _Colin Barker_, Oct 14 2020
%o A283812 (Python) def A283812(n): return 720*7**n - 1800*6**n + 1560*5**n - 540*4**n + 62*3**n - 2**n # _Indranil Ghosh_, Mar 17 2017
%Y A283812 Row 6 of array A099594.
%K A283812 nonn,easy
%O A283812 0,2
%A A283812 _Seiichi Manyama_, Mar 17 2017