A027646 Denominators of poly-Bernoulli numbers B_n^(k) with k=3.

1, 8, 216, 288, 54000, 7200, 3704400, 35280, 7938000, 16800, 78262800, 304920, 4923832914000, 535392, 32464832400, 240240, 265832869302000, 2082880800, 50052680603125200, 387017631, 186286292470278000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• K. Imatomi, M. Kaneko, E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
• M. Kaneko, Poly-Bernoulli numbers
• Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A027645.

Programs

Magma

A027646:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027646(n,3): n in [0..20]]; // G. C. Greubel, Aug 02 2022

Maple

a:= (n, k)-> denom( (-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m=0..n)):
seq(a(n, 3), n = 0..30);

Mathematica

With[{k=3}, Table[Sum[(-1)^(n+j)*j!*StirlingS2[n,j]*(j+1)^(-k), {j,0,n}], {n,0, 40}]]//Denominator (* G. C. Greubel, Aug 02 2022 *)

SageMath

def A027646(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027646(n,3) for n in (0..20)] # G. C. Greubel, Aug 02 2022

Formula

a(n) = denominator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 3.