Dobinski-type relations and the log-normal distribution

We consider sequences of generalized Bell numbers B(n), n 1, 2, ..., which can be represented by Dobinski-type summation formulae, i.e. B(n)1/Ck0[P(k)]n/D(k), with P(k) a polynomial, D(k) a function of k and C const. They include the standard Bell numbers (P(k) k, D(k) k!,Ce), their generalizations Br,r(n), r 2, 3, ..., appearing in the normal ordering of powers of boson monomials (P(k) (k+r)!/k!, D(k) k!, Ce), variants of 'ordered' Bell numbers Bo(p)(n) (P(k) k, D(k) (p+1/p)k, C 1 + p, p 1, 2 ...), etc. We demonstrate that for , , , t positive integers (, t 0), [B(n2 + n + )]t is the nth moment of a positive function on (0,) which is a weighted infinite sum of log-normal distributions