  The aim of this article is to define some new families of the special
numbers. These numbers provide some further motivation for computation of
combinatorial sums involving binomial coefficients and the Euler kind numbers
of negative order. We can show that these numbers are related to the well-known
numbers and polynomials such as the Stirling numbers of the second kind and the
central factorial numbers, the array polynomials, the rook numbers and
polynomials, the Bernstein basis functions and others. In order to derive our
new identities and relations for these numbers, we use a technique including
the generating functions and functional equations. Finally, we not only give a
computational algorithm for these numbers, but also some numerical values of
these numbers and the Euler numbers of negative order with tables. We also give
some combinatorial interpretations of our new numbers.
