I do not know how to proceed to this question: We have in integer partitions: $Φ(x) = 1/ P(x) = \prod_{k\geq 1}(1 - x^k)$ This part I perfectly understand. We then see that $Φ(x)=\sum_{n\geq0} (q_e (n) - q_o(n) )x^n$ ( $q_e$ partition into even parts and $q_o$ partition into odd parts), this I also somewhat understand.
We are then given that $Φ(x)=1 + \sum_{k\geq0} (-1)^k (x^{k(3k-1)/2} + x^{k(3k+1)/2})$ and we are asked to write it in a simpler version that is $Φ(x)=\sum_{k}(-1)^k (x^{k(3k-1)/2})$ So I can see that the +1 is taken out and the second part of the equation, and that k is no longer greater or equal to zero, but I have no idea how to prove this.
I would really appreciate your help! Thank you very much!!