Multinomial distribution with negative binomial prior on the number of trials

Let (X0,X1,...,Xk) given n>0 ~ Multinomial (n, p0,p1,...,pk), where
pi's sum to 1.
Let n ~ Negative Binomial (r,p), r>0, integer

What is the unconditional distribution of (X1,...,Xk) if it is known
that X0=r?

I tried to multiply the multinomial and negative binomial p.m.f's and
'integrate out' n, but this does not seem to be straightforward. Is
there any trick?

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