TITLE: Integer partition identity
QUESTION [8 upvotes]: Let be $ p(n,4)$  number of partitions of $ n $ in parts no greater than $4$ then
$$p(13n,4)-p(n,4)={61\over4}n^3 + {35\over2}n^2  + \frac{45+3(-1)^n}{8}n$$ 
I can prove this identity that seems to me very interesting (In my work, still unpublished, I need 3 or 4 pages to explain it at all). If someone has any reference where is mentioned above formula lets me know, or a short proof if it is possible. See [A191698] in OEIS

REPLY [3 votes]: With regard to Qiaochu's answer, the partial fraction decomposition is messy (compared, say, to anything you'd put on a calculus exam), having 8 terms, and coefficients like 59/288. It's worked out in detail in the lovely book by Andrews and Eriksson, Integer Partitions, in Section 6.3, A formula for $p(n,4)$, pages 58-60. They give several formulas for $p(n,4)$, perhaps the simplest is $$p(n,4)={\rm\ nearest\ integer\ to\ }{(n+5)(n^2+n+22+18\left[{n\over2}\right])\over144}$$ 
I suppose one could use this to have a go at $q(n)=p(13n,4)-p(n,4)$. But it might be better to work directly with a generating function for $q(n)$.