The partial Bell polynomials are given by :
$B_{n,k}(x_{1},x_{2},...,x_{n-k+1})=\sum \frac{n!}{j_{1}!j_{2}!...j_{n-k+1}!}\left(\frac{x_{1}}{1!} \right )^{j_{1}}\left(\frac{x_{2}}{2!} \right )^{j_{2}}...\left(\frac{x_{n-k+1}}{(n-k+1)!} \right )^{j_{n-k+1}}$
where the sum is taken over all sequences $j_{1},j_{2}...,j_{n-k+1}$ of non-negative integers such that :
$j_{1}+j_{2}.. = k $
$j_{1}+2j_{2}+3j_{3} ... = n $
my question is about the case when $k=0$ for some $n$ .
obviously the first condition is satisfied if $j_{1}=j_{2}=j_{3} ... = 0$
but then the second condition is violated.
Any insights are more than welcome .