Wolfram says Sondow (2005)[1] and Sondow and Zudilin (2006)[2] noted the inequality: $$\frac{1}{4rm}\left[\frac{(r+1)^{r+1}}{r^r}\right]^m < {(r+1)m \choose m} < \left[\frac{(r+1)^{r+1}}{r^r}\right]^m$$ for $m$ a positive integer and $r\ge 1$ a real number. We can then use $${n+k-1\choose k} < {n+k\choose k} = {(r + 1)m \choose m}$$ with $r = \frac{n}{k}$ and $m = k$. Then we have $${n+k-1\choose k} < \left[\frac{(r+1)^{r+1}}{r^r}\right]^m = \left(\frac{n+k}{k}\right)^{n+k}$$ Now, the binomial expression has the highest value at the middle of the Pascal's triangle. So, in our case, $n+k = 2k$ or at $k = n$. Substituting that in the above inequality, we get: $${n+k-1\choose k} < 2^{2n} = 4^n$$. Therefore, a tighter bound is $${n+k-1\choose k} = O(4^n)$$. You can also see that the lower bound for the maximum value is $${n+k-1\choose k} = \Omega\left(\frac{4^n}{n}\right)$$ References: [1] Sondow, J. "Problem 11132." Amer. Math. Monthly 112, 180, 2005. [2] Sondow, J. and Zudilin, W. "Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper" Ramanujan J. 12, 225-244, 2006.