Consider a sequence $\{b_k\}$ define via: $$ b_k = \sum_{n=0}^\infty \frac{(n+k)!}{n!}a_n. $$ I would like to invert this transform. That is, I would like to know the coefficients $c_{nk}$ such that $$ a_n = \sum_{k=0}^\infty c_{nk} b_k $$ (if they exists).
The following might be helpful. I've found that if I make a small change $$ b_k = \sum_{n=0}^\infty \frac{(n+k)!}{n!}a_{n+k} $$ then $$ a_n=\sum_{k=0}^\infty \frac{(-1)^k}{n!k!} b_{n+k}. $$
Possibly useful: formally (i.e. without worrying about convergence) the Exponential Generating Function of $b_k$ is
$$ \sum_{k=0}^\infty b_k \frac{t^k}{k!} = \sum_{n=0}^\infty\sum_{k=0}^\infty \frac{(n+k)!}{n!k!} a_n t^k = \sum_{n=0}^\infty a_n (1-t)^{-n-1}$$