I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:


I have the homogeneous polynomial function
$$f(X) = \sum\_{u\_1, \dots u\_n = 1}^n X\_{u\_1} \cdots X\_{u\_n}$$
where $n$ is even, and the differential operator
$$ L = \sum\_{j=1}^n \lambda\_j \frac{\partial^2}{\partial X\_j^2},$$
where $\lambda\_j$ are some nonzero numbers.


Problem: Calculate $L^{n/2} f(X)$. Obviously, this is constant and of the form
$$ L^{n/2} f(X) = C(n) \sum\_{u\_1, \dots u\_{n/2}=1}^{n/2} \lambda\_{u\_1} \cdots \lambda\_{u\_{n/2}},$$
for some number $C(n)$. But what is $C(n)$?



Combinatorics: Product Rules.
It seems to me that in either your expression for $L$ or for $L^{n/2}f(X)$, you need to replace $\lambda\_j$ with $1/\lambda\_j$. (I see this has been corrected, so my previous sentence is irrelevant.) In any event, since $f(X)=(X\_1+\cdots+X\_n)^n$ it is clear that $C(n)=n!$. 


