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I can prove this identity that seems to me very interesting (In my work, still unpublished, I need 3 or 4 pages to explain it at all).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