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This arises as a distribution of 'infected' species in some population (stationary distribution in an MC)

$$ p(x) = c(n)x a(n)^x $$ where $a(n) >1$ and $c(n)$ is such that $\sum_{x} p(x)=1$.

Both questions with the same title don't answer it, I checked.

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Assuming $x$ is intended to run over positive integers, you mean $a(n) < 1$, and this is a (shifted) Negative Binomial distribution with shape parameter $2$. –  Robert Israel Feb 16 '12 at 23:22
$x$ is defined over a finite set of positive integers, bit $a(n) >1$. –  sigma.z.1980 Feb 16 '12 at 23:55

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