  We consider summation of some finite and infinite functional p-adic series
with factorials. In particular, we are interested in the infinite series which
are convergent for all primes p, and have the same integer value for an integer
argument. In this paper, we present rather large class of such p-adic
functional series with integer coefficients which contain factorials. By
recurrence relations, we constructed sequence of polynomials A_k(n;x) which are
a generator for a few other sequences also relevant to some problems in number
theory and combinatorics.
