A027643 - OEIS

A027643

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

1, 1, -1, -1, 7, 1, -38, -5, 11, 7, -3263, -15, 13399637, 7601, -8364, -91, 1437423473, 3617, -177451280177, -745739, 166416763419, 3317609, -17730427802974, -5981591, 51257173898346323, 5436374093, -107154672791057, -213827575

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..521

K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5

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.

FORMULA

a(n) = numerator of Sum_{k=0..n} W(n,k)*h(k+1) with W(n,k) = (-1)^(n-k)*k!* Stirling2(n+1,k+1) the Worpitzky numbers and h(n) = Sum_{k=1..n} 1/k^2 the generalized harmonic numbers of order 2. - Peter Luschny, Sep 28 2017

MAPLE

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

seq(a(n), n=0..27);

MATHEMATICA

k=2; Table[Numerator[(-1)^n Sum[(-1)^m m! StirlingS2[n, m]/(m+1)^k, {m, 0, n}]], {n, 0, 27}] (* Michael De Vlieger, Oct 28 2015 *)

PROG

(Magma)

A027643:= func< n, k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;

[A027643(n, 2): n in [0..30]]; // G. C. Greubel, Aug 02 2022

(SageMath)

def A027643(n, k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n, j)/(j+1)^k for j in (0..n)) )

[A027643(n, 2) for n in (0..30)] # G. C. Greubel, Aug 02 2022