n! / n^n

[Leroy Quet]

rlahaye at new.rr.com wrote:
>There are a number of sequences in the OEIS that are related to Stirling's 
>approximation for the factorial function.  Among them A055775 gives the 
>floor of n^n/n! and A073225 gives the ceiling.
>....

Related. Perhaps a comment should be added to sequences A055775 and 
A073225 that
n^n/n! = A001142(n)/A001142(n-1), where A001142(n) is
product{k=0 to n} C(n,k) (where C() is a binomial coefficent).


>As the subject line of this e-mail indicates, I'm interested in the sum of 
>n!/n^n, i.e. Sum[n=1 to infinity, n!/n^n].  I *believe* this converges to 
>very roughly 1.879853...If so, is it at all interesting...?  And if it is, 
>perhaps it's decimal expansion could be considered for addition to the 
>OEIS ala pi, e etc...?  From Stirling it should approximate to 
>(sqrt(2*pi*n))/e^n.  I realize there is probably nothing of interest here, 
>but am just curious.
>
>Ross

I can answer, perhaps, a related question,

what is sum{m=1 to oo} m^m y^m/m!.

(Converges for |y|<1/e.)

 From my sci.math post at:

http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&oe=ISO-8859-1&safe=off&t
hreadm=b4be2fdf.0308201622.257d34ac%40posting.google.com&rnum=35&prev=

I get (if I did not err)

sum{m=1 to oo} x^m  m^m exp(-x m) /m! =

x/(1-x).

So, 

letting y = x/e^x,  ie. x = -W(-y), we get

sum{m=1 to oo} m^m y^m /m!

=  -W(-y)/(1+W(-y)).

(W() is {one branch of} the Lambert W-function, of course.)


thanks,
Leroy Quet