i'm looking at this question.
"$n$ is positive integer, $T(n)$ is the number of the partitions with odd parts of $n$ integer.
Show that $T(n) ≡ 0 \text{ (mod } 2)$ if $n \neq \dfrac{k(3k ∓ 1)}{2}.$"
i know about this question that the numbers of type $\dfrac{k(3k∓1)}{2}$ is called Pentagonal Numbers.But I don't know how to do it. May you help me please?
The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. By the Pentagonal number theorem, the excess of the number of paritions of $n$ into an even number of distinct parts over those with an odd number of distinct parts is $\pm1$ if $n$ is a pentagonal number and $0$ otherwise. It follows that the parity of the number of partitions of $n$ into distinct parts (and hence into odd parts) is odd if $n$ is a pentagonal number and even otherwise.