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

[Joseph Biberstine]

The example you gave has an off-by-one row index as follows:

Exp[1] = (exp. weighted sum of column n)/(exp. weighted sum of row n+1)

Note that the row index is incremented.  Mathematica simplifies this
fraction to LaguerreL[-(n+1), 1]/LaguerreL[n+1, -1].  This is not equal
to Exp[1], but interestingly seems to approach Exp[1] monotonically from
below as n approaches infinity.

I'm sure you meant:

Exp[1] = (exp. weighted sum of column n)/(exp. weighted sum of row n)

Mathematica simplifies this to LaguerreL[-(n+1), 1]/LaguerreL[n, -1].
Although I can't get it to explicitly resolve that this is exactly
Exp[1], the values do seem to bear this truth out.

So, I believe your example should read: e.g. for n=3,

    1/0! + 4/1! + 10/2! + 20/3! + 35/4! + ...  weighted col-sum
e = -----------------------------------------  -------------------
           1/0! + 3/1! + 3/2! + 1/3!           weighted row-sum

This does seem to hold for every col/row pair in the triangle.

Nice find :)
-JRB

Gottfried Helms 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
>