Maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ Is there an expression for the maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ (i.e. $\max_{k\in\{0,\ldots,n\}}{n\choose k}\lambda^k)$ in terms of elementary functions of $n$ and $\lambda$?
This seems like a simple calculus problem but my usual method, finding the zero of the derivative, doesn't work here since $n \choose k$ is not differentiable.

In discrete case, it may be useful to look at ratio of successful terms.  Here, let $a_k = \binom{n}k \lambda^k$.  Then:
$$\frac{a_{k+1}}{a_k} = \lambda\frac{n-k}{k+1}$$
As $k$ increases from $1$ to $n$, it is easily seen that the numerator decreases and the denominator increases, so the fraction decreases steadily from $\frac{n-1}2$ to $0$.  At some point it becomes less than $1$, and the term before that point would be the maximum.  So we solve for
$$\lambda(n-k) \le k+1 \implies k \ge \frac{n\lambda-1}{\lambda+1}$$
