While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define $h_j(X)$ for $j \ge 0$ by $h_0(X) = 1$ and $$ h_{j}(X) = \frac{X + 1}{j}\left(- X \frac{\mathrm{d}}{\mathrm{d}\ X} + j\right)h_{j-1}(X)$$ for $j \ge 1$.

I was interested in these because $h_j(X)$ is the unique polynomial of degree $j$ such that $$\left(\frac{t}{e^t - 1}\right)^{j+1} \cdot h_j(e^t - 1) = 1 + O(t^{j+1}),$$ and in fact it follows from the recurrence that $$\left(\frac{t}{e^t - 1}\right)^{j+1} \cdot h_j(e^t - 1) = 1 + (-1)^j \sum_{n \ge j+1} \binom{n-1}{j} \frac{B_n t^n}{n!}$$ where $B_n$ are the usual Bernoulli numbers.

Now, I can't believe that these polynomials $h_j$ aren't some terribly classical well-studied thing, but they don't match any of the standard sequences of polynomials I could find on the web. Does anyone recognise these?