A027647 Numerators of poly-Bernoulli numbers B_n^(k) with k=4.

1, 1, -49, 41, 26291, -1921, 845233, 1048349, -60517579, -50233, 506605371959, 823605863, -53797712101337483, -7784082036337, 8049010408144441, 246319059461, -3910018782537447618421, 1090400590625849

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• frac

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..387
• 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. A027648.

Programs

Magma

A027647:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027647(n,4): n in [0..20]]; // 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, 4), n = 0..30);

Mathematica

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 *)

SageMath

def A027647(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027647(n,4) for n in (0..20)] # 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 = 4.