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A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.

%I A027648 #27 Aug 04 2022 05:14:35
%S A027648 1,16,1296,3456,3240000,144000,1555848000,59270400,5000940000,1587600,
%T A027648 9762501672000,11269843200,221794053611130000,39390663312000,
%U A027648 5849513501832000,519437318400,407131014322092060000,1063331477208000
%N A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.
%H A027648 Seiichi Manyama, <a href="/A027648/b027648.txt">Table of n, a(n) for n = 0..807</a>
%H A027648 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 A027648 M. Kaneko, <a href="https://www.emis.de/journals/JTNB/1997-1/kaneko.ps">Poly-Bernoulli numbers</a>
%H A027648 Masanobu Kaneko, <a href="https://doi.org/10.5802/jtnb.197">Poly-Bernoulli numbers</a>, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
%H A027648 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%F A027648 a(n) = denominator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 4.
%p A027648 a:= (n, k) -> denom( (-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m = 0..n) );
%p A027648 seq(a(n, 4), n = 0..30);
%t A027648 With[{k = 4}, Table[Denominator@ Sum[((-1)^(m + n))*m!*StirlingS2[n, m]*(m + 1)^(-k), {m, 0, n}], {n, 0, 17}]] (* _Michael De Vlieger_, Mar 18 2017 *)
%o A027648 (Magma)
%o A027648 A027648:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
%o A027648 [A027648(n,4): n in [0..20]]; // _G. C. Greubel_, Aug 02 2022
%o A027648 (SageMath)
%o A027648 def A027648(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
%o A027648 [A027648(n,4) for n in (0..20)] # _G. C. Greubel_, Aug 02 2022
%Y A027648 Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027649, A027650, A027651.
%K A027648 nonn,frac,easy
%O A027648 0,2
%A A027648 _N. J. A. Sloane_