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

[Gottfried Helms]

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