I'd like to show that:
The number of partitions of $n$ such that parts appear 2,3 or 5 times is equal to the number of partitions of $n$ into parts congruent to $\pm 2$, $\pm 3$, $6$ mod $12$.
The generating function for partitions of n sucht that parts appear 2,3 or 5 times is $\prod_{k=1}^{\infty}(1+q^{2k}+q^{3k}+q^{5k})= \prod_{k \geq1} (1+q^k)(1+q^{2k})(1-q^k+q^{2k})= \prod_{k\geq1}\frac{(1-q^{2k})(1-q^{4k})(1-q^k+q^{2k})}{(1-q^k)(1-q^{2k})}= \prod_{k\geq0}\frac{(1-q^{12k+4})(1-q^{12k+8})(1-q^{12(k+1)})(1-q^k+q^{2k})}{(1-q^{12k+1})...(1-q^{12k+12})}$
A couple of things then cancel but I can't get rid of the $(1-q^k+q^{2k})$ term. Any help would be appreciated.