OFFSET
0,2
COMMENTS
Triangle A209518 * [1, -1/3, 1/18, 1/90, ...] = [1, 0, 0, 0, 0, ...]. - Gary W. Adamson, Mar 09 2012
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Daniel Berhanu, Hunduma Legesse, Arithmetical properties of hypergeometric bernoulli numbers, Indagationes Mathematicae, 2016.
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.
F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.
FORMULA
Given a variant of Pascal's triangle (cf. A209518) in which the two rightmost diagonals are deleted, invert the triangle and extract the leftmost column. Considered as a sequence, we obtain A006568/A006569: (1, -1/3, 1/18, 1/90, ...). - Gary W. Adamson, Mar 09 2012
EXAMPLE
a(0), a(1), a(2), ... = (1, -1/3, 1/18, ...) = leftmost column of the inverse of the 3 X 3 matrix [1; 1, 3; 1, 4, 6; ...].
MATHEMATICA
rows = 28; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
M[[All, 1]] // Denominator (* Jean-François Alcover, Jul 14 2018 *)
PROG
(Sage)
def A006568_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).denominator())
return R
print(A006568_list(28)) # Peter Luschny, Feb 20 2016
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
EXTENSIONS
More terms from Peter Luschny, Feb 20 2016
STATUS
approved