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%I #23 Aug 02 2022 20:28:03
%S 1,4,36,24,450,40,2205,168,350,120,38115,88,40990950,10920,5005,24,
%T 130180050,136,1935088155,3192,177827650,1320,1539340803,184,
%U 304521767550,10920,37182145,24,2814316555050,1160
%N Denominators of poly-Bernoulli numbers B_n^(k) with k=2.
%H Seiichi Manyama, <a href="/A027644/b027644.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 M. Kaneko, <a href="http://ftp.linux.cz/mount/muni.cz/EMIS/journals/JTNB/1997-1/kaneko.ps">Poly-Bernoulli numbers</a>
%H 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 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F a(n) = denominator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 2.
%p a := n -> denom(add((-1)^(n-m)*m!*Stirling2(n, m)/(m+1)^2, m=0..n)):
%p seq(a(n), n = 0..29);
%t f[n_]:= (-1)^n*Sum[(-1)^m*m!*StirlingS2[n, m]/(m + 1)^2, {m, 0, n}];
%t Table[Denominator[f[n]], {n, 0, 30}] (* _Robert G. Wilson v_, Oct 28 2004 *)
%o (Magma)
%o A027644:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
%o [A027644(n,2): n in [0..30]]; // _G. C. Greubel_, Aug 02 2022
%o (SageMath)
%o def A027644(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
%o [A027644(n,2) for n in (0..30)] # _G. C. Greubel_, Aug 02 2022
%Y Cf. A027643.
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_
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