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A141459 A027760(n)/2 for n>=1, a(0) = 1. 7
1, 1, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365, 1, 3, 1, 255, 1, 399, 1, 165, 1, 69, 1, 1365, 1, 3, 1, 435, 1, 7161, 1, 255, 1, 3, 1, 959595, 1, 3, 1, 6765, 1, 903, 1, 345, 1, 141, 1, 23205, 1, 33, 1, 795, 1, 399, 1, 435, 1, 177, 1, 28393365, 1, 3, 1, 255, 1, 32361, 1, 15, 1, 2343, 1, 70050435 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Conjecture: a(n) = denominator of integral_{0..1}(log(1-1/x)^n) dx. - Jean-François Alcover, Feb 01 2013

Define the generalized Bernoulli function as B(s,z) = -s*z^s*HurwitzZeta(1-s,1/z) for Re(1/z) > 0 and B(0,z) = 1 for all z; further the generalized Bernoulli polynomials as Bp(m,n,z) = Sum_{j=0..n} B(j,m)*C(n,j)*(z-1)^(n-j) then the a(n) are denominators of Bp(2,n,1), i. e. of the generalized Bernoulli numbers in the case m=2. The numerators of these numbers are A157779(n). - Peter Luschny, May 17 2015

LINKS

Table of n, a(n) for n=0..72.

Peter Luschny, Generalized Bernoulli Numbers and Polynomials

FORMULA

a(2*n+1) = 1. a(2*n)= A001897(n).

a(n) = denominator(0^n + Sum_{j=1..n} zeta(1-j)*(2^j-2)*j*C(n,j)). - Peter Luschny, May 17 2015

EXAMPLE

The denominators of 1, 0, -1/3, 0, 7/15, 0, -31/21, 0, 127/15, 0, -2555/33, 0, 1414477/1365, ...

MAPLE

Bfun := (s, z) -> `if`(s=0, 1, -s*z^s*Zeta(0, 1-s, 1/z): # generalized Bernoulli function

Bpoly := (m, n, z) -> add(Bfun(j, m)*binomial(n, j)*(z-1)^(n-j), j=0..n): # generalized Bernoulli polynomials

seq(Bpoly(2, n, 1), n=0..50): denom([%]);

# which simplifies to:

a := n -> 0^n+add(Zeta(1-j)*(2^j-2)*j*binomial(n, j), j=1..n):

seq(denom(a(n)), n=0..50); # Peter Luschny, May 17 2015

MATHEMATICA

a[n_] := If[OddQ[n], 1, Denominator[-2*(2^(n - 1) - 1)*BernoulliB[n]]]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jan 30 2013 *)

CROSSREFS

Cf. A027760, A141459, A157779, A226157.

Sequence in context: A101820 A055301 A214073 * A176727 A080924 A232179

Adjacent sequences:  A141456 A141457 A141458 * A141460 A141461 A141462

KEYWORD

nonn

AUTHOR

Paul Curtz, Aug 08 2008

EXTENSIONS

Prepended 1 and set offset to 0 by Peter Luschny, May 17 2015

STATUS

approved

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Last modified August 27 19:53 EDT 2015. Contains 261098 sequences.