Explain which partitions of $n$ does the coefficient of $x^n$ count in the following...

Question

$\frac{1}{1-x}\frac{1}{1-x^3}(1+x^2+x^4)$. I've expanded out the 2 fractions with Taylor series then found that it equals $1+\sum_{n=1}^{\infty}nx^n$ but I'm not sure if this is right/ what I can say from this. Any help would be appreciated.

Answer 1

From the product, it counts the number of partitions that use:

• any number of 1's
• 0, 1, or 2 of 2's
• any number of 3's
• no other terms.