login
Poly-Bernoulli numbers B_n^(k) with k = -6.
3

%I #31 May 18 2024 14:54:17

%S 1,64,1394,20266,237686,2441314,22934774,202229266,1701740006,

%T 13821281314,109214866454,844558486066,6419351203526,48118995192514,

%U 356641942834934,2618939805811666,19085432672558246,138206899494338914,995563711729120214,7139963278111582066,51017526215427244166

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

%C a(n) is also the number of acyclic orientations of the complete bipartite graph K_{6,n}. - _Vincent Pilaud_, Sep 16 2020

%H Seiichi Manyama, <a href="/A283812/b283812.txt">Table of n, a(n) for n = 0..1179</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (27,-295,1665,-5104,8028,-5040).

%F a(n) = 720*7^n - 1800*6^n + 1560*5^n - 540*4^n + 62*3^n - 2^n.

%F From _Colin Barker_, Oct 14 2020: (Start)

%F 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 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 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 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 (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 (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 (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 Row 6 of array A099594.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Mar 17 2017