A006568 Denominators of generalized Bernoulli numbers. (Formerly M3046)

1, 3, 18, 90, 270, 1134, 5670, 2430, 7290, 133650, 112266, 1990170, 9950850, 2296350, 984150, 117113850, 351341550, 33657930, 21597171750, 3410079750, 572893398, 33613643250, 834229509750, 108812544750, 544062723750, 18280507518, 105464466450, 18690647109750

Offset

0,2

Comments

Triangle A209518 * [1, -1/3, 1/18, 1/90, ...] = [1, 0, 0, 0, 0, ...]. - Gary W. Adamson, Mar 09 2012

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..27.

Daniel Berhanu, Hunduma Legesse, Arithmetical properties of hypergeometric bernoulli numbers, Indagationes Mathematicae, 2016.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568.

Abdul Hassen and Hieu D. Nguyen, Hypergeometric Zeta Functions, arXiv:math/0509637 [math.NT], Sep 27 2005.

F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.

Index entries for sequences related to Bernoulli numbers.

Formula

Given a variant of Pascal's triangle (cf. A209518) in which the two rightmost diagonals are deleted, invert the triangle and extract the leftmost column. Considered as a sequence, we obtain A006568/A006569: (1, -1/3, 1/18, 1/90, ...). - Gary W. Adamson, Mar 09 2012

Example

a(0), a(1), a(2), ... = (1, -1/3, 1/18, ...) = leftmost column of the inverse of the 3 X 3 matrix [1; 1, 3; 1, 4, 6; ...].

Mathematica

rows = 28; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
M[[All, 1]] // Denominator (* Jean-François Alcover, Jul 14 2018 *)

Prog

(Sage)
def A006568_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+2)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).denominator())
    return R
print(A006568_list(28)) # Peter Luschny, Feb 20 2016

Crossrefs

Cf. A006569, A132092-A132099, A209518.

Sequence in context: A037295 A290331 A124811 * A181955 A147518 A088336

Adjacent sequences: A006565 A006566 A006567 * A006569 A006570 A006571

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

More terms from Peter Luschny, Feb 20 2016

Status

approved