Is there a closed for the coefficient of $x^n$ in $(1+x)(1+x^2)(1+x^3)\cdots$? If not, then what is the closest to a closed form that anyone has found? (An infinite series that approximates it perhaps? Or maybe something of the type "the coefficient of $x^n$ is the closest integer to ____" where the blank is filled by a complicated looking expression? Etc.)
Note: I already know that that the coefficient of $x^n$ can be interpreted as the number of ways to write $n$ as the sum of zero or one $1$'s, zero or one $2$'s, zero or one $3$'s, etc., but I haven't been able to prove anything interesting from this view point.
Any help is appreciated, thanks.
Number of partitions of $n$ into distinct parts = number of partitions of $n$ into odd parts. This is a very well-studied sequence. See OEIS sequence A000009, in particular the Formula section and the References and Links.