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A006568 Denominators of generalized Bernoulli numbers.

%I A006568 M3046 #52 Jan 24 2024 08:49:11
%S A006568 1,3,18,90,270,1134,5670,2430,7290,133650,112266,1990170,9950850,
%T A006568 2296350,984150,117113850,351341550,33657930,21597171750,3410079750,
%U A006568 572893398,33613643250,834229509750,108812544750,544062723750,18280507518,105464466450,18690647109750
%N A006568 Denominators of generalized Bernoulli numbers.
%C A006568 Triangle A209518 * [1, -1/3, 1/18, 1/90, ...] = [1, 0, 0, 0, 0, ...]. - _Gary W. Adamson_, Mar 09 2012
%D A006568 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006568 Daniel Berhanu, Hunduma Legesse, <a href="http://dx.doi.org/10.1016/j.indag.2016.11.014">Arithmetical properties of hypergeometric bernoulli numbers</a>, Indagationes Mathematicae, 2016.
%H A006568 Hector Blandin and Rafael Diaz, <a href="http://arXiv.org/abs/0708.0809">Compositional Bernoulli numbers</a>, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568.
%H A006568 Abdul Hassen and Hieu D. Nguyen, <a href="http://arXiv.org/abs/math/0509637">Hypergeometric Zeta Functions</a>, arXiv:math/0509637 [math.NT], Sep 27 2005.
%H A006568 F. T. Howard, <a href="http://dx.doi.org/10.1215/S0012-7094-67-03465-5">A sequence of numbers related to the exponential function</a>, Duke Math. J., 34 (1967), 599-615.
%H A006568 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A006568 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
%e A006568 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; ...].
%t A006568 rows = 28; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
%t A006568 M[[All, 1]] // Denominator (* _Jean-François Alcover_, Jul 14 2018 *)
%o A006568 (Sage)
%o A006568 def A006568_list(len):
%o A006568     f, R, C = 1, [1], [1]+[0]*(len-1)
%o A006568     for n in (1..len-1):
%o A006568         f *= n
%o A006568         for k in range(n, 0, -1):
%o A006568             C[k] = C[k-1] / (k+2)
%o A006568         C[0] = -sum(C[k] for k in (1..n))
%o A006568         R.append((C[0]*f).denominator())
%o A006568     return R
%o A006568 print(A006568_list(28)) # _Peter Luschny_, Feb 20 2016
%Y A006568 Cf. A006569, A132092-A132099, A209518.
%K A006568 nonn,frac
%O A006568 0,2
%A A006568 _N. J. A. Sloane_
%E A006568 More terms from _Peter Luschny_, Feb 20 2016