# Generating function for 3 -core partitions: Part II

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