A027648 - OEIS

%I #27 Aug 04 2022 05:14:35

%S 1,16,1296,3456,3240000,144000,1555848000,59270400,5000940000,1587600,

%T 9762501672000,11269843200,221794053611130000,39390663312000,

%U 5849513501832000,519437318400,407131014322092060000,1063331477208000

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

%H Seiichi Manyama, <a href="/A027648/b027648.txt">Table of n, a(n) for n = 0..807</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 M. Kaneko, <a href="https://www.emis.de/journals/JTNB/1997-1/kaneko.ps">Poly-Bernoulli numbers</a>

%H 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 <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 = 4.

%p a:= (n, k) -> denom( (-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[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 (Magma)

%o A027648:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;

%o [A027648(n,4): n in [0..20]]; // _G. C. Greubel_, Aug 02 2022

%o (SageMath)

%o 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(n,4) for n in (0..20)] # _G. C. Greubel_, Aug 02 2022

%Y Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027649, A027650, A027651.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_