prove this equality 
First, I try to use the Taylor series but it doesn't work well.
And someone said I can use fundamental theorem of calculus but I don't see it.
can anyone give me a good hit?
thanks

Here is a start:
Let
$f_n( x)
=D_n(x^{n-1}e^{1/x})
$,
where $D_n$ 
is the $n^{th}$
derivative.
By Leibnitz's formula
$D_n(a(x)b(x))
=\sum_{k=0}^n \binom{n}{ k} D_k(a(x))D_{n-k}(b(x)) 
$.
(This is readily proved by induction
using the regular product rule
for derivatives.)
Then get the appropriate derivatives.
Note that many of the derivatives
may be zero,
which might simplify things.
