%I #49 Aug 04 2022 05:11:41
%S 1,8,46,230,1066,4718,20266,85310,354106,1455278,5938186,24104990,
%T 97478746,393095438,1581931306,6356390270,25511588986,102304505198,
%U 409992599626,1642294397150,6576150108826,26325519044558,105364834103146,421647614381630,1687155299822266
%N Poly-Bernoulli numbers B_n^(k) with k=-3.
%C a(n) is also the number of acyclic orientations of the complete bipartite graph K_{3,n}. - _Vincent Pilaud_, Sep 15 2020
%H Vincenzo Librandi, <a href="/A027650/b027650.txt">Table of n, a(n) for n = 0..500</a>
%H K. Imatomi, M. Kaneko, and E. Takeda, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Kaneko/kaneko2.html">Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values</a>, J. Int. Seq. 17 (2014) # 14.4.5.
%H Ken Kamano, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Kamano/kamano2.html">Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers</a>, J. Int. Seq. 13 (2010), 10.5.2.
%H Ken Kamano, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kamano/kamano5.html">Sums of Products of Poly-Bernoulli Numbers of Negative Index</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
%H Masanobu Kaneko, <a href="http://www.numdam.org/item?id=JTNB_1997__9_1_221_0">Poly-Bernoulli numbers</a>, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
%H Takao Komatsu, <a href="https://doi.org/10.22436/jnsa.012.12.05">Some recurrence relations of poly-Cauchy numbers</a>, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-26,24).
%F a(n) = 6*4^n - 6*3^n + 2^n. - _Vladeta Jovovic_, Nov 14 2003
%F G.f.: (1-x)/((1-2*x)*(1-3*x)*(1-4*x)).
%F E.g.f.: exp(2*x) - 6*exp(3*x) + 6*exp(4*x). - _G. C. Greubel_, Aug 02 2022
%p a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n, j)/(j+1)^k, j=0..n);
%p seq(a(n, -3), n = 0..30);
%t Table[6*4^n-6*3^n+2^n, {n,0,30}] (* _G. C. Greubel_, Feb 07 2018 *)
%o (Magma) [6*4^n-6*3^n+2^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 17 2011
%o (PARI) Vec((1-x)/((1-2*x)*(1-3*x)*(1-4*x)) + O(x^30)) \\ _Michel Marcus_, Feb 13 2015
%o (SageMath) [2^n -6*3^n +6*4^n for n in (0..30)] # _G. C. Greubel_, Aug 02 2022
%Y Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027651.
%Y First differences of A016269.
%Y Row 3 of array A099594.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_