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 A006568 Denominators of generalized Bernoulli numbers. (Formerly M3046) 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568. Daniel Berhanu, Hunduma Legesse, Arithmetical properties of hypergeometric bernoulli numbers, Indagationes Mathematicae, 2016. 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. 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-A132106, A209518 Sequence in context: A037295 A290331 A124811 * A181955 A147518 A088336 Adjacent sequences:  A006565 A006566 A006567 * A006569 A006570 A006571 KEYWORD nonn,frac AUTHOR EXTENSIONS More terms from Peter Luschny, Feb 20 2016 STATUS approved

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Last modified August 11 05:42 EDT 2020. Contains 336422 sequences. (Running on oeis4.)