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%I #52 Aug 04 2022 05:11:59
%S 1,16,146,1066,6902,41506,237686,1315666,7107302,37712866,197451926,
%T 1023358066,5262831302,26903268226,136887643766,693968021266,
%U 3508093140902,17693879415586,89084256837206,447884338361266,2249284754708102,11285908565322946,56587579617416246
%N Poly-Bernoulli numbers B_n^(k) with k=-4.
%C a(n) is also the number of acyclic orientations of the complete bipartite graph K_{4,n}. - _Vincent Pilaud_, Sep 16 2020
%H Vincenzo Librandi, <a href="/A027651/b027651.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 Hiroyuki Komaki, <a href="http://arxiv.org/abs/1503.04933">Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index</a>, arXiv:1503.04933 [math.NT], 2015.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (14,-71,154,-120).
%F a(n) = 24*5^n -36*4^n +14*3^n -2^n. - _Vladeta Jovovic_, Nov 14 2003
%F G.f.: (1+4*x)*(1-x)^2/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
%F E.g.f.: 24*exp(5*x) - 36*exp(4*x) + 14*exp(3*x) - exp(2*x). - _G. C. Greubel_, Feb 07 2018
%p a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n,j)/(j+1)^k, j=0..n);
%p seq(a(n, -4), n=0..30);
%t Table[24*5^n -36*4^n +14*3^n -2^n, {n,0,30}] (* _G. C. Greubel_, Feb 07 2018 *)
%t LinearRecurrence[{14,-71,154,-120},{1,16,146,1066},30] (* _Harvey P. Dale_, Nov 20 2019 *)
%o (Magma) [24*5^n-36*4^n+14*3^n-2^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 17 2011
%o (PARI) Vec((1+4*x)*((1-x)^2)/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) + O(x^30)) \\ _Michel Marcus_, Feb 13 2015
%o (SageMath) [24*5^n -36*4^n +14*3^n -2^n for n in (0..30)] # _G. C. Greubel_, Aug 02 2022
%Y Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027650.
%Y Row n=4 of array A099594.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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