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A027646 Denominators of poly-Bernoulli numbers B_n^(k) with k=3.

%I A027646 #20 Aug 02 2022 20:29:56
%S A027646 1,8,216,288,54000,7200,3704400,35280,7938000,16800,78262800,304920,
%T A027646 4923832914000,535392,32464832400,240240,265832869302000,2082880800,
%U A027646 50052680603125200,387017631,186286292470278000
%N A027646 Denominators of poly-Bernoulli numbers B_n^(k) with k=3.
%H A027646 Seiichi Manyama, <a href="/A027646/b027646.txt">Table of n, a(n) for n = 0..1000</a>
%H A027646 K. Imatomi, M. Kaneko, 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 A027646 M. Kaneko, <a href="http://ftp.linux.cz/mount/muni.cz/EMIS/journals/JTNB/1997-1/kaneko.ps">Poly-Bernoulli numbers</a>
%H A027646 Masanobu Kaneko, <a href="http://jtnb.cedram.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 A027646 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A027646 a(n) = denominator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 3.
%p A027646 a:= (n, k)-> denom( (-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m=0..n)):
%p A027646 seq(a(n, 3), n = 0..30);
%t A027646 With[{k=3}, Table[Sum[(-1)^(n+j)*j!*StirlingS2[n,j]*(j+1)^(-k), {j,0,n}], {n,0, 40}]]//Denominator (* _G. C. Greubel_, Aug 02 2022 *)
%o A027646 (Magma)
%o A027646 A027646:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
%o A027646 [A027646(n,3): n in [0..20]]; // _G. C. Greubel_, Aug 02 2022
%o A027646 (SageMath)
%o A027646 def A027646(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
%o A027646 [A027646(n,3) for n in (0..20)] # _G. C. Greubel_, Aug 02 2022
%Y A027646 Cf. A027645.
%K A027646 nonn,frac
%O A027646 0,2
%A A027646 _N. J. A. Sloane_