# A question on partial Bell polynomials

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 .

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$B_{n,0}=0$ unless $n=0$. $B_{0,0}=1$.