Let \(n,s\) be positive integers, and let \(r = 1 + \frac{1}{s}\). In this note we prove that if the sequence \(\{a_n(r)\}_{n=1}^{\infty}\) satisfies \(ra_n(r)= \sum_{k=1}^{n}\binom{n}{k}a_k(r), n> 1\), then \(a_n(r) = na_1(r)\left[(n -1)! / {(s+1)}(log r)^n+{{1/r(s+1)}} \right]\). Moreover, we obtain a related combinatorial identity.