Curious binomial-identity /A002720 (small correction of prev. post)

[Jonathan Post]

There are several formulae for A002720, including a(n) = Sum k!C(n, k)^2,
k=0..n.

What you write might be true in the asymptotic limit, but not for any term,
as each term is rational and dividing by e would make each term
transcendental.  I'm sure that you meant the limit, right?

On 11/25/06, Gottfried Helms <Annette.Warlich at t-online.de> wrote:
>
> Am 25.11.2006 20:26 schrieb Gottfried Helms:
> > By chance I came across this curious identity
> > involving the Pascal-triangle.
> >
> > Assume a row n, say n=4 and the column n, combined
> > each weighted with the running factorial as in the example:
> >
> >
> >            1/0! + 4/1! + 10/2! + 20/3! + 35/4! + ...  weighted col-sum
> >  ratio =
> -----------------------------------------  -------------------
> >            1/0! + 4/1! + 6/2! + 4/3! + 1/4!           weighted row-sum
> >
> >
> > then
> >
> >   ratio = e  (=exp(1))
> >
>
> ------------------------------------
>
> > The actual sums are the entries of A002720
> >  http://www.research.att.com/~njas/sequences/A002720
> >
> that should be corrected; the entries in A002720 are
>
> A(n)  =  weighted-rowsum(n) * n!
>        =  weighted-colsum(n) * n! / exp(1)
>
> I forgot to mention the additional n!, since in numerator and
> denominator of the above fraction they cancel out, sorry.
>
> Gottfried Helms
>
>
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