  For the calculation of Springer numbers (of root systems) of type $B_n$ and
$D_n$, Arnold introduced a signed analogue of alternating permutations, called
$\beta_n$-snakes, and derived recurrence relations for enumerating the
$\beta_n$-snakes starting with $k$. The results are presented in the form of
double triangular arrays ($v_{n,k}$) of integers, $1\le |k|\le n$. An Arnold
family is a sequence of sets of such objects as $\beta_n$-snakes that are
counted by $(v_{n,k})$. As a refinement of Arnold's result, we give analogous
arrays of polynomials, defined by recurrence, for the calculation of the
polynomials associated with successive derivatives of $\tan x$ and $\sec x$,
established by Hoffman. Moreover, we provide some new Arnold families of
combinatorial objects that realize the polynomial arrays, which are signed
variants of Andr\'{e} permutations and Simsun permutations.
