Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length 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? $$\sum_{n\geq1}O_3(n)\,q^n=(1+q)\sum_{k\geq1}q^{k^2}.$$


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 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 $$\sum_{n\geq1}O_3(n)\,q^n=\sum_{k\geq1}(q^{k^2}+q^{k^2+1}).$$


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