Integer-valued polynomials on prime numbers and logarithm power expansion

We prove that two sequences arising from two different domains are equal. The first one, { d ( n ) } n ∈ N , comes from the following power expansion: ( − ln ( 1 − x ) x ) m = ( ∑ k = 1 + ∞ x k k + 1 ) m = ∑ n = 0 ∞ B n ( m ) d ( n ) x n where B n ( X ) is a primitive polynomial of Z [ X ] . The second sequence, { e ( n ) } n ∈ N , is the factorial sequence of the set of prime numbers or, equivalently, e ( n ) is the denominator of the polynomials of degree ≤ n + 1 that take integral values for all prime numbers.