The statement referenced by Igor Rivin http://www.math.clemson.edu/~janoski/ResearchStatement.pdf uses the phrase

> Computationally looking at p(n) we see that for n ≥ 26 the partition function is log-concave [2].

I had seen this reference before probably about the same time this research statement was first released, and I am skeptical for two reasons.

1.  The phrasing "Computationally..." would seem to indicate some type of calculation.  This cannot involve a computer since it would have to hold for all n larger than 26, and I am not aware of any simplification that allows one to only consider a finite number of cases.  It would have been helpful to at least expound on the type of computations involved.

2.  I checked for the promised reference, and indeed I found it on the CV of the author, http://www.math.clemson.edu/~janoski/VitaTex.pdf, but it refers to the quote below.  I did a quick google search and I could find no reference or anything pointing to a publication.
>Brian Bowers, Neil Calkin, Kerry Gannon, Janine E. Janoski, Katie Joes, Anna Kirkpatrick, The Log Concavity of the Partition Function, (in preparation)


3.  Asymptotics will not provide the answer here, since n sufficiently large doesn't hold up unless you can provide a concrete n and test everything less than it, and I don't believe the Hardy-Ramanujan asymptotic expansion yields any guaranteed error estimates.

4.  It may be possible to use DH Lehmer's estimates to obtain a proof.  In two papers (1937 and 1939) he investigated the coefficients of both the Hardy-Ramanujan asymptotic expansion and the Hardy-Ramanujan-Rademacher expansion.  He provided *guaranteed error bounds* on the remainder terms in the asymptotic expansions so that, for example, his Theorem 13 says that for n>600, only $2/3 \sqrt n$ terms of the Hardy-Ramanujan asymptotic series are needed to estimate p(n) to the nearest integer.

At present, I don't believe the matter is completely settled, despite the overwhelming computational evidence.