  We consider two types of polynomials $F_n (x) = \sum_{\nu=1}^n \nu!
S_2(n,\nu) x^\nu$ and $\hat{F}_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) H_\nu
x^\nu$, where $S_2(n,\nu)$ are the Stirling numbers of the second kind and
$H_\nu$ are the harmonic numbers. We show some properties and relations between
these polynomials. Especially, the identity $\hat{F}_n (-\tfrac{1}{2}) = -
(n-1)/2 \cdot F_{n-1} (-\tfrac{1}{2})$ is established for even $n$, where the
values are connected with Genocchi numbers. For odd $n$ the value of $\hat{F}_n
(-\tfrac{1}{2})$ is given by a convolution of these numbers. Subsequently, we
discuss some of these convolutions, which are connected with Miki type
convolutions of Bernoulli and Genocchi numbers, and derive some 2-adic
valuations of them.
