  The main purpose of this paper is to construct new families of special
numbers with their generating functions. These numbers are related to the many
well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the
Lucas numbers, the Stirling numbers of the second kind and the central
factorial numbers. Our other inspiration of this paper is related to the
Golombek's problem \cite{golombek} \textquotedblleft Aufgabe 1088, El. Math. 49
(1994) 126-127\textquotedblright . Our first numbers are not only related to
the Golombek's problem, but also computation of the negative order Euler
numbers. We compute a few values of the numbers which are given by some tables.
We give some applications in Probability and Statistics. That is, special
values of mathematical expectation in the binomial distribution and the
Bernstein polynomials give us the value of our numbers. Taking derivative of
our generating functions, we give partial differential equations and also
functional equations. By using these equations, we derive recurrence relations
and some formulas of our numbers. Moreover, we give two algorithms for
computation our numbers. We also give some combinatorial applications, further
remarks on our numbers and their generating functions.
