I assume you mean picking a random $k$-subset from {1,...,n}, i.e. drawing without replacement? In this case one finds for $X_j=\alpha_j-j$ that

$$\mathbb{E}(t^{X_j})= {k!\,(n-k)! \over n!} [x^{n-k}]{1 \over (1-tx)^j(1-x)^{k+1-j}}$$

and $$\mu(t):=\mathbb{E} \sum_{j=1}^k t^{X_j}={n+1 \over n+1-k}{1-t^{n+1-k} \over 1-t}- t\,{1-t^{n-k} \over 1-t} -1$$

I assume you mean picking a random $k$-subset from {1,...,n}, i.e. drawing without replacement? In this case one finds for $X_j=\alpha_j-j$ that

$$\mathbb{E}(t^{X_j})= {k!\,(n-k)! \over n!} [x^{n-k}]{1 \over (1-tx)^j(1-x)^{k+1-j}}$$

and $$\mu(t):=\mathbb{E} \sum_{j=1}^k t^{X_j}={n+1 \over n+1-k}{1-t^{n+1-k} \over 1-t}- t\,{1-t^{n-k} \over 1-t} -1={k \over n-k+1}{1-t^{n-k+1} \over 1-t}$$