From this old paper (Glasser 1962, https://www.jstor.org/stable/1266576) I discovered that
$$\frac{x^{[i]}}{n^i}$$
is an unbiased estimator for $\lambda^i$, where $x = \sum_1^n x_j$ for a set of $n$ samples $x_j$ from the Poisson distribution with parameter $\lambda$, and where $x^{[i]}$ is defined as
$$x^{[i]} = \frac{x!}{(x-i)!}$$
In other words, given $x_j$, we can construct unbiased estimators for all (integer) powers of $\lambda$.
My question is this: is there an equivalent set of estimators for the binomial distribution? I.e. can we estimate all powers of, say, $pN$, where $p$ is the success probability and $N$ is the number of trials? I tested out the same formula numerically, since I figured that the Poisson distribution is just a limit of the binomial distribution so perhaps it would still work, but it does not. Numerically it seems that $x^{[i]}/n^i$ becomes an unbiased estimator of $(pN)^i$ only in the limit of large $N$.
But can this be fixed somehow? Is there a similar set of estimators that work in the general binomial case?