A note on the Borwein conjecture

A conjecture of Borwein asserts that for any positive integers $n$ and $k$, the coefficient $a_{3k}$ of $q^{3k}$ in the expansion of $\prod_{j=0}^n (1-q^{3j+1})(1-q^{3j+2})$isnonnegative\mathrm{.}Inthispaperweprovethatforany$0 \leq k\leq n$,thereisaconstant$0<c<1$suchthat$$a_{3k}+a_{3(n+1)+3k}+\cdots+a_{3n(n+1)+3k}=\frac {2\cdot 3^{n}} {n+1}(1+O(c^n)).$$Inparticular,$$a_{3k}+a_{3(n+1)+3k}+\cdots+a_{3n(n+1)+3k}>0.$$