Curious binomial-identity /A002720

[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

Regards -

Gottfried Helms