The multiset coefficient
$$\left (\binom{n}{k} \right ) = \binom{n+k-1}{k}$$
denotes the number of multisets of cardinality k, with elements taken from a finite set of cardinality n. I'm interested in how this number grows when both n and k are increased by an integer x.
$$\left (\binom{n+x}{k+x} \right ) = \binom{n+k+2x-1}{k+x}$$
I've done some plotting and it appears this number grows exponentially with x, but I haven't been able to prove this. How would I do it?
The ratio between consecutive terms
$$\frac{\binom{n + k + 2x + 1}{k + x + 1}}{\binom{n + k + 2x - 1}{k + x}} = \frac{(n + k + 2x + 1)(n + k + 2x)}{(k + x + 1)(n + x)}$$
converges to $4$ as $x \to \infty$.