\magnification=1200
\hsize=4in
\nopagenumbers
\noindent
{\bf Michael E. Hoffman}
\medskip
\noindent
{\bf Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences}
\vskip.5cm


Let $P_n$ and $Q_n$ be the polynomials obtained by repeated differentiation
of the tangent and secant functions respectively.  From the exponential
generating functions of these polynomials we develop relations among their
values, which are then applied to various numerical sequences which occur
as values of the $P_n$ and $Q_n$.  For example, $P_n(0)$ and
$Q_n(0)$ are respectively the $n$th tangent and secant numbers, while
$P_n(0)+Q_n(0)$ is the $n$th Andr\'e number.  The Andr\'e numbers,
along with the numbers $Q_n(1)$ and $P_n(1)-Q_n(1)$, are the Springer
numbers of root systems of types $A_n$, $B_n$, and $D_n$
respectively, or alternatively (following V. I. Arnol'd) count the number
of ``snakes'' of these types.  We prove this for the latter two cases
using combinatorial arguments.  We relate the values of $P_n$ and $Q_n$
at $\sqrt3$ to certain ``generalized Euler and class numbers'' of D. Shanks,
which have a combinatorial interpretation in terms of 3-signed
permutations as defined by R. Ehrenborg and M. A. Readdy.  Finally,
we express the
values of Euler polynomials at any rational argument in terms of $P_n$
and $Q_n$, and from this deduce formulas for Springer and Shanks numbers
in terms of Euler polynomials.


\bye