cp's OEIS Frontend

A027645 Numerators of poly-Bernoulli numbers B_n^(k) with k=3.

%I A027645 #25 Aug 02 2022 20:30:37
%S A027645 1,1,-11,-1,1243,-49,-75613,599,234671,-803,-4955857,53443,
%T A027645 921931911863,-449291,-23461249769,1237447,917870505450709,
%U A027645 -82659252107,-959539811053709101,145633840717,20593004175300735901,-12278015226517
%N A027645 Numerators of poly-Bernoulli numbers B_n^(k) with k=3.
%H A027645 Seiichi Manyama, <a href="/A027645/b027645.txt">Table of n, a(n) for n = 0..443</a>
%H A027645 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 A027645 M. Kaneko, <a href="http://ftp.linux.cz/mount/muni.cz/EMIS/journals/JTNB/1997-1/kaneko.ps">Poly-Bernoulli numbers</a>.
%H A027645 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), pp. 221-228.
%H A027645 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%F A027645 a(n) = numerator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 3.
%p A027645 a:= (n, k)-> numer((-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m=0..n)):
%p A027645 seq(a(n, 3), n = 0..30);
%t A027645 With[{k=3}, Table[Sum[(-1)^(n+j)*j!*StirlingS2[n,j]*(j+1)^(-k), {j,0,n}], {n, 0, 40}]]//Numerator (* _G. C. Greubel_, Aug 02 2022 *)
%o A027645 (Magma)
%o A027645 A027645:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
%o A027645 [A027645(n,3): n in [0..30]]; // _G. C. Greubel_, Aug 02 2022
%o A027645 (SageMath)
%o A027645 def A027645(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
%o A027645 [A027645(n,3) for n in (0..30)] # _G. C. Greubel_, Aug 02 2022
%Y A027645 Cf. A027646.
%K A027645 sign,frac
%O A027645 0,3
%A A027645 _N. J. A. Sloane_