A027644 Denominators of poly-Bernoulli numbers B_n^(k) with k=2.

1, 4, 36, 24, 450, 40, 2205, 168, 350, 120, 38115, 88, 40990950, 10920, 5005, 24, 130180050, 136, 1935088155, 3192, 177827650, 1320, 1539340803, 184, 304521767550, 10920, 37182145, 24, 2814316555050, 1160

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. A027643.

Programs

Magma

A027644:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027644(n,2): n in [0..30]]; // G. C. Greubel, Aug 02 2022

Maple

a := n -> denom(add((-1)^(n-m)*m!*Stirling2(n, m)/(m+1)^2, m=0..n)):
seq(a(n), n = 0..29);

Mathematica

f[n_]:= (-1)^n*Sum[(-1)^m*m!*StirlingS2[n, m]/(m + 1)^2, {m, 0, n}];
Table[Denominator[f[n]], {n, 0, 30}] (* Robert G. Wilson v, Oct 28 2004 *)

SageMath

def A027644(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027644(n,2) for n in (0..30)] # 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 = 2.