A050925 Numerator of (n+1)*Bernoulli(n).

1, -1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0, -26315271553053477373, 0, 38089920879940267, 0

Offset: 0

Author

N. J. A. Sloane, Dec 30 1999

Keywords

• sign
• easy
• frac
• nice

Comments

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

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

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

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..200
• M. Kaneko, A recurrence formula for the Bernoulli numbers, Proc. Japan Acad., 71 A (1995), 192-193.
• OEIS Wiki, Autosequence
• S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A050932, A000367/A002445, A002882, A003245, A127187, A127188, A242179, A106831, A000142, A123125.

Programs

Haskell

a050925 n = a050925_list !! n
a050925_list = 1 : -1 : (tail $ map (numerator . sum) $
   zipWith (zipWith (%))
   (zipWith (map . (*)) (drop 2 a000142_list) a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014

Mathematica

Numerator[Table[(n+1)BernoulliB[n],{n,0,40}]] (* Harvey P. Dale, May 13 2012 *)

PARI

a(n)=numerator(bernfrac(n)*(n+1)) \\ Charles R Greathouse IV, Feb 07 2017