%I A006569 M3731
%S A006569 1,1,1,1,1,5,1,7,13,307,479,1837,100921,15587,23737,5729723,14731223,9129833,
%T A006569 2722952839,4700745901,1556262845,190717213397,24684889339847,50242799489,
%U A006569 148437433077277,8592042383621,221844330989749,176585172615885307,9245931549625447
%V A006569 1,-1,1,1,-1,-5,-1,7,13,-307,-479,1837,100921,15587,-23737,-5729723,14731223,
               9129833,
%W A006569 2722952839,-4700745901,-1556262845,190717213397,24684889339847,-50242799489,
%X A006569 -148437433077277,-8592042383621,221844330989749,176585172615885307,-9245931549625447
%N A006569 Numerators of generalized Bernoulli numbers.
%D A006569 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006569 F. T. Howard, A sequence of numbers related to the exponential function, 
               Duke Math. J., 34 (1967), 599-615.
%H A006569 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%H A006569 Hector Blandin and Rafael Diaz, <a href="http://arXiv.org/pdf/0708.0809">
               Compositional Bernoulli numbers</a>, Page 7, 2nd table is identical 
               to A006569/A006568.
%H A006569 Abdul Hassen and Hieu D. Nguyen, <a href="http://arXiv.org/pdf/math/0509637">
               Hypergeometric Zeta Functions</a>, Sep 27 2005.
%F A006569 Recurrence relation: bernoulli(n+1)=a[n+1]-sum(binomial(n+1, r)*bernoulli(r)*a[n+2-r], 
               r=1..n+1); a[0]=1. (p. 603 of the Howard reference). - Deutsch - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 23 2005
%F A006569 E.g.f. for fractions: x^2/2 / (e^x-1-x). [From Franklin T. Adams-Watters 
               (FrankTAW(AT)Netscape.net), Nov 04 2009]
%p A006569 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); (Deutsch)
%Y A006569 Cf. A006568.
%Y A006569 Cf. A132092-A132106.
%Y A006569 Sequence in context: A100122 A001945 A051854 this_sequence A126155 A021197 
               A073116
%Y A006569 Adjacent sequences: A006566 A006567 A006568 this_sequence A006570 A006571 
               A006572
%K A006569 frac,sign,easy
%O A006569 0,6
%A A006569 N. J. A. Sloane (njas(AT)research.att.com).
%E A006569 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 23 2005