(08/31/2010, 08:04 AM)mike3 Wrote: Doing some tests, it appears that

has only one occurrence of , and no higher powers of it, and it never seems to be multiplied by any sort of n-dependent coefficient. This means that . I don't have a proof at this point, ...

Now I've got that proof. Here it goes:

We have

(note how the connection to the Faà di Bruno's formula is clear)

with the sum taken over all sequences of non-negative integers such that and , and is the sum of the . As should be obvious from the formula, we see that the value in only occurs in the component . What kind of occurrences of are possible there? Using the formulas for the , we see that , which means that all but one must be zero, and that one that is must be 1. The second constraint, , would imply that if all but one are zero, the nonzero one must equal where is its position in the sequence. This would mean must divide . The first constraint, though, said it must be 1, which means and so the only possible sequence of is ( terms). This means , which is obviously just . So contains only one term with , which is just itself. And then follows trivially.