OFFSET
0,6
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
frac,sign,easy
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Jan 23 2005
STATUS
approved