The following email, written by Daniel and posted on Doron's webpage, came to my attention this morning:
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/DanielKaneMessage.txt
It shows very succinctly that the exponent of the growth of the
coefficients in Wilf's 6th problem is:
(6n)^{1/3} log(n)/3
I'm happy to see that Daniel made this observation.
I'm also pleased to report that Jim Fill, Svante Janson, and I (working together) independently made the same observation in November 2011, which we chose to write as a function of 6n, very similarly:
(6 n)^{1/3} \ln[(6 n)^{1/3}]
We haven't yet published our result, and indeed, we would like to know more. For instance, is it possible to get a first-order expression for not just the exponent, but for the growth of the coefficients themselves?
As Doron points out in his arxiv preprint:
http://arxiv.org/pdf/1201.4093v1.pdf
and as Jim, Svante, and I discussed, it would probably be a result akin to the classic Hardy and Ramanujan (and, later, Rademacher refinement) of the asymptotic growth of the number of integer partitions.
I don't know if you are interested in pushing further on this problem, but I certainly am, and all would be welcome to join.
I am also copying Maciej Ireneusz Wilczynski, who derived the first 500+ terms (I had submitted 300 terms to Sloane last year, but I'm impressed that Maciej went way beyond this). I think he is also interested in a precise analysis of the asymptotics.
Best wishes to all,
Mark