%I #37 Aug 04 2022 05:14:58
%S 1,1,-1,-1,7,1,-38,-5,11,7,-3263,-15,13399637,7601,-8364,-91,
%T 1437423473,3617,-177451280177,-745739,166416763419,3317609,
%U -17730427802974,-5981591,51257173898346323,5436374093,-107154672791057,-213827575
%N Numerators of poly-Bernoulli numbers B_n^(k) with k=2.
%H Seiichi Manyama, <a href="/A027643/b027643.txt">Table of n, a(n) for n = 0..521</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 Masanobu Kaneko, <a href="https://www.emis.de/journals/JTNB/1997-1/kaneko.ps">Poly-Bernoulli numbers</a>, Journal de Théorie des Nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
%H Masanobu Kaneko, <a href="http://dx.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 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%F a(n) = numerator of Sum_{k=0..n} W(n,k)*h(k+1) with W(n,k) = (-1)^(n-k)*k!* Stirling2(n+1,k+1) the Worpitzky numbers and h(n) = Sum_{k=1..n} 1/k^2 the generalized harmonic numbers of order 2. - _Peter Luschny_, Sep 28 2017
%p a := n -> numer((-1)^n*add( (-1)^m*m!*Stirling2(n, m)/(m+1)^2, m=0..n)):
%p seq(a(n), n=0..27);
%t k=2; Table[Numerator[(-1)^n Sum[(-1)^m m! StirlingS2[n, m]/(m+1)^k, {m, 0, n}]], {n, 0, 27}] (* _Michael De Vlieger_, Oct 28 2015 *)
%o (Magma)
%o A027643:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
%o [A027643(n,2): n in [0..30]]; // _G. C. Greubel_, Aug 02 2022
%o (SageMath)
%o def A027643(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
%o [A027643(n,2) for n in (0..30)] # _G. C. Greubel_, Aug 02 2022
%Y Cf. A027641, A027642, A027644, A027645, A027646, A027647, A027648, A027649, A027650, A027651.
%K sign,frac
%O 0,5
%A _N. J. A. Sloane_