We give a simple proof of the moment-indeterminacy on the half-line of the sequence ( n ! ) t for t > 2 , using Lin’s condition. Under a logarithmic self-decomposability assumption, the method conveys to power moment sequences defined as the rising factorials of a given Bernstein function, and to more general infinitely divisible moment sequences. We also provide a very short proof of the infinite divisibility of all the integer moment sequences recently investigated in [16], including Fuss–Catalan’s.