[seqfan] Re: A005002 inconsistent with wikipedia entry about Stirling numbers of the second kind?

[franktaw at netscape.net]

They are the same.  The formula defining A005000 does count
what A006505 is counting.

Two points:

The offset of A005000 is wrong.  From the definition, it should
be 0, which matches A006505.

The definition of A005000 is overly elaborate.  The polynomials
g_n are not needed (although they are of some independent
interest and should perhaps be added as a table).  We can just
set a(0) = 1, and
a(n) = Sum_{k=0}^{n-3} C(n-1,k)*a(k).

(Even keeping the polynomials, there is no need to special
case any n > 0.  The cases n=1 and n=2 are the normal
interpretation of an empty sum, and n=3, 4, and 5 are
perfectly normal.)

Franklin T. Adams-Watters

-----Original Message-----
From: Richard Mathar <mathar at strw.leidenuniv.nl>

It ought be possible to show by s.o. with some flexibility in
converting e.g.f's to binomial sums to demonstrate that
(apart from offset) A005000 and A006505 are the same -- this is
just proposed by comparing all the numbers.

Richard