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{d(n)}n∈N, comes from the following power expansion: (−ln(1−x)x)m=(∑k=1+∞xkk+1)m=∑n=0∞Bn(m)d(n)xn where Bn(X)Bn(X) is a primitive polynomial of Z[X]Z[X]. The second sequence, {e(n)}n∈N{e(n)}n∈N, is the factorial sequence of the set of prime numbers or, equivalently, e(n)e(n) is the denominator of the polynomials of degree ≤n+1≤n+1 that take integral values for all prime numbers.