A006568 Denominators of generalized Bernoulli numbers.
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
Examples
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; ...].
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- 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.
Programs
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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 *) -
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
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
Extensions
More terms from Peter Luschny, Feb 20 2016
Comments