  By studying Cameron's operator in terms of determinants, two kinds of
"integer" sequences of incomplete numbers were introduced. One was the sequence
of restricted numbers, including $s$-step Fibonacci sequences. Another was the
sequence of associated numbers, including Lam\'e sequences of higher order. By
the classical Trudi's formula and the inverse relation, more expressions were
able to be obtained. These relations and identities can be extended to those of
sequence of negative integers or rational numbers. As applications, we consider
hypergeometric Bernoulli, Cauchy and Euler numbers with some modifications.
