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 distinct parts and let $d\_t(n)$ be the number of such partitions that are $t$-cores. Then it is easy to see $d\_2(n)=c\_2(n)$.



> 
> **QUESTION.** Is this true?
> $$\sum\_{n\geq0}d\_3(n)\,q^n=\sum\_{k\geq0}q^{k^2}
> +\sum\_{k\geq1}q^{2\binom{k+1}2}.$$
> 
> 
> 


Note that I have simplified the generating function from
$$\frac12\prod\_{n\geq1}(1-q^{2n})(1+q^{2n-1})^2+\prod\_{n\geq1}(1-q^{2n})(1+q^{2n})^2-\frac12.$$



Generating function for $3$-core partitions
The set of $3$-core partitions can be described explicitly. 


**Theorem** The partition $\lambda=\{\lambda\_1,\lambda\_2,\dots\}$ of length $k$ (that is, $\lambda\_k > 0$ but $\lambda\_{k+1} = \lambda\_{k+2} = \cdots = 0$) is a $3$-core if and only if the sequence of differences $\{\lambda\_1-\lambda\_2,\lambda\_2-\lambda\_3,\dots,\lambda\_k - \lambda\_{k+1}\}$ is of the form $\{2,2,\dots,2,1,0,1,0,\dots ,1\}$ or $\{2,2,\dots,2,0,1,0,1,\dots ,1\}$.


**Proof:** It is easy to check by hand that a $3$-hook appears in the situations where


* a) some member of the sequence is $\geq 3$,
* b) two members of the sequence in a row are $0$'s,
* c) there is a $1$ in the sequence that is not followed by a $0$,
* d) there is a $0$ in the sequence that is not followed by a $1$.


These correspond to the four possible shapes of a $3$ hook-strip in the boundary. If all of these patterns are avoided, then the partition has no hooks of length $3$.


Therefore the partitions with distinct parts that are $3$-cores have a difference sequence $\{2,2,\dots,2\}$ or $\{2,2,\dots,2,1\}$. The size of the partitions in the first case are given by $2\binom{k+1}{2}$ and the sizes in the second case are given by $k^2$, where $k\geq 1$, and this implies your generating function.


