A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.

1, 16, 1296, 3456, 3240000, 144000, 1555848000, 59270400, 5000940000, 1587600, 9762501672000, 11269843200, 221794053611130000, 39390663312000, 5849513501832000, 519437318400, 407131014322092060000, 1063331477208000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..807
• K. Imatomi, M. Kaneko, and 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. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027649, A027650, A027651.

Programs

Magma

A027648:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027648(n,4): 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, 4), n = 0..30);

Mathematica

With[{k = 4}, Table[Denominator@ 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 A027648(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027648(n,4) 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 = 4.