[seqfan] Re: A005002 inconsistent with wikipedia entry about Stirling numbers of the second kind?
Richard Mathar
mathar at strw.leidenuniv.nl
Wed Oct 29 15:04:54 CET 2008
It turns out that the corresponding polynomials (coefficients)
are A059022. So in some sense A006505 are the row sums of A059022.
ftaw> From seqfan-bounces at list.seqfan.eu Wed Oct 29 10:28:41 2008
ftaw> To: seqfan at list.seqfan.eu
ftaw> Date: Wed, 29 Oct 2008 05:19:57 -0400
ftaw> From: franktaw at netscape.net
ftaw> Subject: [seqfan] Re: A005002 inconsistent with wikipedia entry about
ftaw> Stirling numbers of the second kind?
ftaw>
ftaw> They are the same. The formula defining A005000 does count
ftaw> what A006505 is counting.
ftaw>
ftaw> Two points:
ftaw>
ftaw> The offset of A005000 is wrong. From the definition, it should
ftaw> be 0, which matches A006505.
ftaw>
ftaw> The definition of A005000 is overly elaborate. The polynomials
ftaw> g_n are not needed (although they are of some independent
ftaw> interest and should perhaps be added as a table). We can just
ftaw> set a(0) = 1, and
ftaw> a(n) = Sum_{k=0}^{n-3} C(n-1,k)*a(k).
ftaw>
ftaw> (Even keeping the polynomials, there is no need to special
ftaw> case any n > 0. The cases n=1 and n=2 are the normal
ftaw> interpretation of an empty sum, and n=3, 4, and 5 are
ftaw> perfectly normal.)
ftaw> ...
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