Consider the function: $$S(n)=\sum_{j=1}^{\infty}\frac{j^n}{2^j}=\frac1 2+\frac{2^n}{4}+\frac{3^n}{8}+\frac{4^n}{16}+\frac{5^n}{32}+...$$
Euler found the sum of the first few of these as:
$S(0)=1$; (as per the usual geometric series.)
$S(1)=2$; etc. This creates a series of sums as follows:
1, 2, 6, 26, 150, 1082, 9366, 94586, … , which is the OEIS sequence A000629, “Number of necklaces of partitions of $n+1$ labelled beads”.
My question is - what connection is there between this infinite series and necklace combinatorics? Or is the connection accidental or illusory?