On a moment problem related to Bernstein functions

We give a simple proof of the moment-indeterminacy of the sequence $(n!{)}^t$ for $t>2,$ using Lin’s condition. Under a logarithmic self-decomposability assumption, the method conveys to power 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 moment sequences recently investigated in Lin (2017), including Fuss-Catalan’s.