Cameron's operators, determinants and Bernoulli numbers

    By studying Cameron's operator in terms of determinants, we introduce two kinds of the sequences of incomplete numbers.  One is the sequence of restricted numbers, which can yield $s$-step Fibonacci sequences in the simplest case. Another is the sequence of associated numbers, which can yield Lam\'e sequences of higher order in the simplest case. 
    By the classical Trudi's formula and the inverse relation, more expressions can 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.