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%I #22 Aug 02 2022 19:04:45
%S 1,1,-49,41,26291,-1921,845233,1048349,-60517579,-50233,506605371959,
%T 823605863,-53797712101337483,-7784082036337,8049010408144441,
%U 246319059461,-3910018782537447618421,1090400590625849
%N Numerators of poly-Bernoulli numbers B_n^(k) with k=4.
%H Seiichi Manyama, <a href="/A027647/b027647.txt">Table of n, a(n) for n = 0..387</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) = numerator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 4.
%p a:= (n, k)-> numer((-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m=0..n)):
%p seq(a(n, 4), n = 0..30);
%t With[{k = 4}, Table[Numerator@ Sum[((-1)^(m + n))*m!*StirlingS2[n, m]*(m + 1)^(-k), {m, 0, n}], {n, 0, 17}]] (* _Michael De Vlieger_, Mar 18 2017 *)
%o (Magma)
%o A027647:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
%o [A027647(n,4): n in [0..20]]; // _G. C. Greubel_, Aug 02 2022
%o (SageMath)
%o def A027647(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
%o [A027647(n,4) for n in (0..20)] # _G. C. Greubel_, Aug 02 2022
%Y Cf. A027648.
%K sign,frac
%O 0,3
%A _N. J. A. Sloane_
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