Dear Neil, I can partially reply to your request.
In his 2002 survey,
MR1898209 (2003c:34001) Ilyashenko, Yu. Centennial history of Hilbert's 16th problem. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 301--354 (electronic). (Reviewer: Lubomir Gavrilov) 34-02 (34C07 37C10 37F75)
the author mentions that Petrovskii and Landis gave an incorrect proof that the Hilbert number H(n) is bounded above by P_3(n), "a certain polynomial of degree 3 [in n]". The reference is to:
MR0073004 (17,364d) Petrovski\u\i, I. G.; Landis, E. M. On the number of limit cycles of the equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are polynomials of 2nd degree. (Russian) Mat. Sb. N.S. 37(79) (1955), 209--250.
No, this paper contains only a proof of H(2) = 3. The proper reference for n>2 is [PL2] Landis, E. M., Petrovskii, I. G.; On the number of limit cycles of the equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are polynomials (Russian). Mat. Sb. N.S. 43(85) (1957), 149-168. MR 19,746c.
My question is, what is this sequence P_3(n) ?
P_3(n) = (6n^3-7n^2-11n+16)/2 for odd n and P_3(n) = (6n^3-7n^2+n+4)/2 for even n. Moreover, L&P point out that by Otrokov, 1954, there exist equations with (n^2+5n-14)/2 limit cycles (hopefully this result is correct, unlike their upper bound P_3(n).)
Later in the article Ilyashenko mentions a second Hilbert-type sequence E(n) (on page 305). Same question.
I know nothing. Valery Liskovets