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A164555 Numerators of the "original" Bernoulli numbers. 80
1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

Apart from a sign flip in a(1), the same as A027641.

a(n) is also the numerator of the n-th term of the Binomial transform of the sequence of Bernoulli numbers, i.e., of the sequence of fractions A027641(n)/A027642(n).

a(n)/A027642(n) with e.g.f. x/(1-exp(-x)) is the a-sequence for the Sheffer matrix A094645, see the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jun 20 2011

a(n)/A027642(n) give also the row sums of the rational triangle of the coefficients of the Bernoulli polynomials A053382/A053383 (falling powers) or A196838/A196839 (rising powers). - Wolfdieter Lang, Oct 25 2011

Given M = the beheaded Pascal's triangle, A074909; with B_n as a vector V, with numerators shown: (1, 1, 1,...). Then M*V = [1, 2, 3, 4, 5,...]. If the sign in a(1) is negative in V, then M*V = [1, 0, 0, 0,...]. - Gary W. Adamson, Mar 09 2012

One might interpret the term ""original" Bernoulli numbers" as numbers given by the e.g.f. x/(1-exp(-x)). - Peter Luschny, Jun 17 2012

LINKS

Table of n, a(n) for n=0..40.

Tom Rike, Sums of powers and Bernoulli numbers.

MAPLE

A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; fi; end: # R. J. Mathar, Aug 26 2009

seq(numer(n!*coeff(series(t/(1-exp(-t)), t, n+2), t, n)), n=0..40); # Peter Luschny, Jun 17 2012

MATHEMATICA

CoefficientList[ Series[ x/(1 - Exp[-x]), {x, 0, 40}], x]*Range[0, 40]! // Numerator (* Jean-François Alcover, Mar 04 2013 *)

PROG

(Haskell)

a164555 n = a164555_list !! n

a164555_list = 1 : map (numerator . sum) (zipWith (zipWith (%))

   (zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)

-- Reinhard Zumkeller, Jul 04 2014

CROSSREFS

Cf. A027641, A027642, A006232, A053382, A053383, A074909, A094645, A196838, A196839.

Cf. A242179, A106831, A000142.

Sequence in context: A036946 A027641 * A176327 A226156 A215616 A129205

Adjacent sequences:  A164552 A164553 A164554 * A164556 A164557 A164558

KEYWORD

sign,frac

AUTHOR

Paul Curtz, Aug 15 2009

EXTENSIONS

Edited and extended by R. J. Mathar, Sep 03 2009

STATUS

approved

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Last modified October 25 23:29 EDT 2014. Contains 248566 sequences.