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

1, 1, -11, -1, 1243, -49, -75613, 599, 234671, -803, -4955857, 53443, 921931911863, -449291, -23461249769, 1237447, 917870505450709, -82659252107, -959539811053709101, 145633840717, 20593004175300735901, -12278015226517

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• frac

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..443
• 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), pp. 221-228.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A027646.

Programs

Magma

A027645:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027645(n,3): n in [0..30]]; // G. C. Greubel, Aug 02 2022

Maple

a:= (n, k)-> numer((-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}]]//Numerator (* G. C. Greubel, Aug 02 2022 *)

SageMath

def A027645(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027645(n,3) for n in (0..30)] # G. C. Greubel, Aug 02 2022

Formula

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