A006569 Numerators of generalized Bernoulli numbers.

1, -1, 1, 1, -1, -5, -1, 7, 13, -307, -479, 1837, 100921, 15587, -23737, -5729723, 14731223, 9129833, 2722952839, -4700745901, -1556262845, 190717213397, 24684889339847, -50242799489, -148437433077277, -8592042383621, 221844330989749, 176585172615885307, -9245931549625447

Offset: 0

Author

N. J. A. Sloane

Keywords

• frac
• sign
• easy

References

• F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568.
• Abdul Hassen and Hieu D. Nguyen, Hypergeometric Zeta Functions, arXiv:math/0509637 [math.NT], Sep 27 2005.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A006568.

Cf. A132092-A132099.

Programs

Maple

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

Mathematica

rows = 29; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
M[[All, 1]] // Numerator (* Jean-François Alcover, Jul 14 2018 *)

Sage

def A006569_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+2)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).numerator())
    return R
print(A006569_list(29)) # Peter Luschny, Feb 20 2016

Formula

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

E.g.f. for fractions: x^2/2 / (e^x-1-x). - Franklin T. Adams-Watters, Nov 04 2009

Extensions

More terms from Emeric Deutsch, Jan 23 2005