Partitions of an integer where each summand appears at most four times

Question

Find the generating function for the number of partitions of an integer (greater than zero), where each summand appears at most four times.

Is it the following?

Answer 1

Well, the usual generating function for the number $p(n)$ of partitions of an integer $n$ is given by :

$$\prod_{n=1}^{\infty}(1+x^n+x^{2n}+x^{3n}+...)=\prod_{n=1}^{\infty}\frac{1}{1-x^n} $$

Following the same argument (there is something missing in the formula you have given), the generating function for the number of partitions of an integer where each summand appears at most four times is :

$$\prod_{n=1}^{\infty}(1+x^n+x^{2n}+x^{3n}+x^{4n})=$$

$$(1+x+x^2+x^3+x^4)(1+x^2+x^{4}+x^{6}+x^{8})(1+x^3+x^{6}+x^{9}+x^{12})...$$