The main purpose of this paper is to construct new families of special
numbers with their generating functions. These numbers are related to many
well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas
numbers, Stirling numbers of the second kind and central factorial numbers.
Our other inspiration of this paper is related to the Golombek's problem
[15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". 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 tables. We give some applications in probability and
statistics. That is, special values of mathematical expectation of 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 come up with a conjecture with two open questions related to
our new numbers. We give two algorithms for computation of our numbers. We
also give some combinatorial applications, further remarks on our new
numbers and their generating functions.