I am currently reading "Introduction to Percolation Theory" by Stauffer and Aharony and am doing the problems. Question 2.2 wants me to calculate a closed-form expression for the $k$-th moment $M_k$ of the cluster number $n_k$ in one dimension. Specifically, they ask me to show that $M_k=\Gamma_k(1-p)^{1-k}$ and to explicitly calculate the $\Gamma_k$
The relevant formulae are:
$M_k = \sum_{s=1}^{\infty}s^kn_s$
$n_s = p^s(1-p)^2$
where $s$ is the cluster size and $p$ is the probability that a site is occupied.
I have gotten this far:
$M_k = \sum_{s=1}^{\infty}s^kp^s(1-p)^2$
$M_k = (1-p)^2\sum_{s=1}^{\infty}s^kp^s$
$M_k = (1-p)^2(p\frac{d}{dp})^k\sum_{s=1}^{\infty}p^s$
$M_k = (1-p)^2(p\frac{d}{dp})^k\frac{p}{1-p}$
At this point, I got stuck because I cannot find a pattern for the derivatives. Plugging into Wolfram Alpha, I get this for the first few:
$(p\frac{d}{dp})\frac{p}{1-p} = \frac{p}{(1-p)^2}$
$(p\frac{d}{dp})^2\frac{p}{1-p} = \frac{p^2+p}{(1-p)^3}$
$(p\frac{d}{dp})^3\frac{p}{1-p} = \frac{p^3+4p^2+p}{(1-p)^4}$
$(p\frac{d}{dp})^4\frac{p}{1-p} = \frac{p^4+11p^3+11p^2+p}{(1-p)^5}$
$(p\frac{d}{dp})^5\frac{p}{1-p} = \frac{p^5+26p^4+56p^3+26p^2+p}{(1-p)^6}$
So it looks like there is some kind of Pascal's Triangle thing going on where the coefficients of the numerators are symmetric, but I am unsure how to proceed from this point.
