Let $\lambda$ be an integer partition: $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq0)$. Further, let $h\_u$ denote the [hook-length](https://en.wikipedia.org/wiki/Hook_length_formula#Example) of the cell $u$.


We call $\lambda$ a $t$-core partition if none of its hooks $h\_u$ equals $t$. Define $c\_t(n)$ to be the number of partitions of $n$ that are $t$-core partitions. It's well-known that 
$$\sum\_{n\geq0}c\_t(n)\,q^n=\prod\_{k=1}^{\infty}\frac{(1-q^{tk})^t}{1-q^k}.$$
For example, $\sum\_{n=0}^{\infty}c\_2(n)\,q^n=\sum\_{k=0}^{\infty}q^{\binom{k+1}2}$.


Now, consider only those partitions of $n$ with **odd parts** and let $O\_t(n)$ be the number of such partitions that are $t$-cores. 



> 
> **QUESTION.** Is this true?
>  $$\sum\_{n\geq1}O\_3(n)\,q^n=(1+q)\sum\_{k\geq1}q^{k^2}.$$
> 
> 
> 



Generating function for 3 -core partitions: Part II
Suppose the partition with $k$ parts $\lambda=\{\lambda\_1\geq \lambda\_2\geq\dots\geq\lambda\_k\geq \lambda\_{k+1}=0\}$ is a partition with only odd parts. Then we have $\lambda\_i-\lambda\_{i+1}$ is even for all $i\leq k-1$ and odd for $i=k$. Using the same characterization I mentioned in the [previous answer](https://mathoverflow.net/questions/312837/generating-function-for-3-core-partitions/312840#312840) we see that the sequence
$\{\lambda\_1-\lambda\_2,\lambda\_2-\lambda\_3,\dots,\lambda\_{k}-\lambda\_{k+1}\}$ for such a $3$-core has to be equal to $\{2,2,\dots,2,1\}$ or $\{2,2,\dots,2,0,1\}$. The first case is a partition with size $k^2$ and the second has size $1+k^2$ which implies the generating function
$$\sum\_{n\geq1}O\_3(n)\,q^n=\sum\_{k\geq1}(q^{k^2}+q^{k^2+1}).$$


