login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A164555 Numerators of the "original" Bernoulli numbers. 97
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

REFERENCES

Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..629

Tom Rike, Sums of powers and Bernoulli numbers.

FORMULA

a(n) = numerator(B(n)) with B(n) = Sum_{k=0..n} (-1)^(n-k) * C(n+1, k+1) * S(n+k, k) / C(n+k, k) and S the Stirling set numbers. - Peter Luschny, Jun 25 2016

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-Fran├žois 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 A249737

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 10 11:31 EST 2016. Contains 279001 sequences.