[seqfan] Re: Identities for the Riemann zeta function gives same first 10 values as A090582

Richard Mathar mathar at strw.leidenuniv.nl
Tue Dec 16 13:09:05 CET 2008

jvp> From seqfan-bounces at list.seqfan.eu Tue Dec 16 05:30:34 2008
jvp> Date: Mon, 15 Dec 2008 19:57:59 -0800
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: seqfan at seqfan.eu
jvp> Content-Disposition: inline
jvp> Subject: [seqfan] Identities for the Riemann zeta function gives same first
jvp>         10 values as A090582
jvp> If I enter the start of the sequence of numerators of the polynomials,
jvp> i.e. by rows of the triangle,
jvp> from page 7 of the pdf of
jvp> Identities for the Riemann zeta function
jvp> Authors: Michael O. Rubinstein
jvp> http://arxiv.org/abs/0812.2592
jvp> I get:
jvp> 1,2,-1,6,-6,1,24,-36,14,-1
jvp> and, up to there, it's the same as
jvp> A090582                 Numerator Q(m,n) of probability P(m,n)=Q(m,n)/n^m to see
jvp> each card at least once if m>=n cards are drawn with replacement from
jvp> a deck of n cards, written in a two-dimensional array read by
jvp> antidiagonals with Q(m,m) as first row and Q(m,1) as first column.
jvp> Law of small numbers, or do the recurrences match?

The numbers are essentially a signed version of A019538. This is obvious
from eq (4.7) in the Rubinstein paper. The Adamchik comment in A090582
says that the entries are "generated by Stirling numbers of the second kind
multiplied by a factorial"; although this is not the precise statement one
would like to have at that place, it indicates that A090582 is just a version
of A019583 read right to left.  This is incorporated by the comment
"Reflected version of A019538" in A090582.
So in summary, the only new aspect of this particular formula of the
Rubinstein paper might be the checkerboard distribution of signs...

Richard Mathar

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