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?



Is this sequence of polynomials well-known?
The first several are:


$$0! \cdot h\_0(x) = 1$$
$$1! \cdot h\_1(x) = x+1$$
$$2! \cdot h\_2(x) = x^2+3 x+2$$
$$3! \cdot h\_3(x) = x^3+7 x^2+12 x+6$$
$$4! \cdot h\_4(x) = x^4+15 x^3+50 x^2+60 x+24$$


Feeding the sequence $2,3,1,6,12,7,1,24,60$ into the [OEIS](http://www.research.att.com/njas/sequences/) gives the [following page](http://oeis.org/A130850), which contains generating functions, relations, and citations to occurrences of this sequence of polynomials in the literature.


