Solving $(1+x)(1+x^2)(1+x^3)(1+x^4)....(1+x^n)$

Question

I want to calculate distinct/odd partition of a given number n

According to the generating function i know

$$(1+x)(1+x^2)(1+x^3)(1+x^4)....(1+x^n)$$

After solving this the coefficient of $x^n$ will be the total number of distinct partition... But when i solve above expressions it's become extremely lengthy and also errors...

Anyone can tell me a general formula for this equation... So i can easily calculate Or any general formula for getting the total number of distinct partitions..

One thing that i know according Leonhard Euler the

number of total distinct partition = total number of odd partition of a given number

Answer 1

See http://mathworld.wolfram.com/PartitionFunctionQ.html

In particular, the function grows like $\dfrac{e^{\pi\sqrt{n/3}}}{4\cdot3^{1/4}n^{3/4}} $ and there is a Rademacher-like convergent series.