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%I #31 Feb 17 2021 12:25:50
%S 1,-1,1,0,-1,0,1,0,-3,0,5,0,-691,0,35,0,-3617,0,43867,0,-1222277,0,
%T 854513,0,-1181820455,0,76977927,0,-23749461029,0,8615841276005,0,
%U -84802531453387,0,90219075042845,0,-26315271553053477373,0,38089920879940267,0
%N Numerator of (n+1)*Bernoulli(n).
%C The denominators are in A050932. The e.g.f. for (n+1)*Bernoulli(n), n >= 0, is (d/dx)(x^2/(exp(x)-1)) = x*(2*(exp(x)-1)- x*exp(x))/(exp(x)-1)^2. - _Wolfdieter Lang_, Jul 15 2013
%C It can be observed that the rational sequence [0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, ...], derived from a(n)/A050932(n), is an autosequence of the first kind. - _Jean-François Alcover_, Jul 21 2017
%C Apparently a(n) = numerator(Sum_{k=0..n-1} (-1)^(n-k+1)*E1(n,k+1)/binomial(n,k+1)) for n >= 2, where E1(n, k) denotes the first-order Eulerian numbers A123125. - _Peter Luschny_, Feb 17 2021
%H N. J. A. Sloane, <a href="/A050925/b050925.txt">Table of n, a(n) for n = 0..200</a>
%H M. Kaneko, <a href="https://doi.org/10.3792/pjaa.71.192">A recurrence formula for the Bernoulli numbers</a>, Proc. Japan Acad., 71 A (1995), 192-193.
%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%H S. C. Woon, <a href="http://www.jstor.org/stable/2691054">A tree for generating Bernoulli numbers</a>, Math. Mag., 70 (1997), 51-56.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%t Numerator[Table[(n+1)BernoulliB[n],{n,0,40}]] (* _Harvey P. Dale_, May 13 2012 *)
%o (Haskell)
%o a050925 n = a050925_list !! n
%o a050925_list = 1 : -1 : (tail $ map (numerator . sum) $
%o zipWith (zipWith (%))
%o (zipWith (map . (*)) (drop 2 a000142_list) a242179_tabf) a106831_tabf)
%o -- _Reinhard Zumkeller_, Jul 04 2014
%o (PARI) a(n)=numerator(bernfrac(n)*(n+1)) \\ _Charles R Greathouse IV_, Feb 07 2017
%Y Cf. A050932, A000367/A002445, A002882, A003245, A127187, A127188, A242179, A106831, A000142, A123125.
%K sign,easy,frac,nice
%O 0,9
%A _N. J. A. Sloane_, Dec 30 1999
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