Expectation of power of difference

Question

Let $X \sim Binomial(M,p)$

Calculate $$ E(X-1)^n, \quad n\geq 0 $$

Answer 1

In principle, these moments can all be extracted from the moment generating function, which for the binomial distribution is

$$\mathbb E[e^{tX}] = (p e^t + 1-p)^M$$

so $$\mathbb E[e^{t(X-1)}] = e^{-t} (p e^t + 1-p)^M $$

Then $$\mathbb E[(X-1)^n] = \left. \dfrac{d^n}{dt^n} E[e^{t(X-1)}] \right|_{t=0} $$

But I don't think there is a closed form.