As you know, the distribution of a negative binomial random variable $X\sim \text{NB}(r,p)$ is: $$P(X=k)={k+r-1\choose k}p^k(1-p)^r$$
Well, by using the general definition of entropy we get that the entropy is expressed by a complicated infinite sum containing binomial coefficients and logarithms.
I was wondering if there is some way to simplify the expression.