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%I M3731 #40 Jan 24 2024 08:49:23
%S 1,-1,1,1,-1,-5,-1,7,13,-307,-479,1837,100921,15587,-23737,-5729723,
%T 14731223,9129833,2722952839,-4700745901,-1556262845,190717213397,
%U 24684889339847,-50242799489,-148437433077277,-8592042383621,221844330989749,176585172615885307,-9245931549625447
%N Numerators of generalized Bernoulli numbers.
%D F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H 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 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 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F Recurrence relation: Bernoulli(n+1) = a(n+1) - Sum_{r=1..n+1} binomial(n+1, r)*Bernoulli(r)*a(n+2-r); a(0)=1 (p. 603 of the Howard reference). - _Emeric Deutsch_, Jan 23 2005
%F E.g.f. for fractions: x^2/2 / (e^x-1-x). - _Franklin T. Adams-Watters_, Nov 04 2009
%p eq:=n->bernoulli(n+1)=a[n+1]-sum(binomial(n+1,r)*bernoulli(r)*a[n+2-r],r=1..n+1): a[0]:=1:for n from 0 to 28 do a[n+1]:=solve(eq(n),a[n+1]) od: seq(numer(a[n]),n=0..29); # _Emeric Deutsch_, Jan 23 2005
%t rows = 29; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
%t M[[All, 1]] // Numerator (* _Jean-François Alcover_, Jul 14 2018 *)
%o (Sage)
%o def A006569_list(len):
%o f, R, C = 1, [1], [1]+[0]*(len-1)
%o for n in (1..len-1):
%o f *= n
%o for k in range(n, 0, -1):
%o C[k] = C[k-1] / (k+2)
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append((C[0]*f).numerator())
%o return R
%o print(A006569_list(29)) # _Peter Luschny_, Feb 20 2016
%Y Cf. A006568.
%Y Cf. A132092-A132099.
%K frac,sign,easy
%O 0,6
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Jan 23 2005