Coefficient of $x^9$ in the Product Find the Coefficient of $x^9$ in $$G(x)=(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots(1+x^{100})$$
My Try:
$$G(x)=P(x)(1+x^{10})(1+x^{11})\cdots(1+x^{100})=P(x)(1+O(x^{10}))$$ Hence Coefficient of $x^9$ in $G(x)$ is Coefficient of $x^9$ in $P(x)$ where $$P(x)=(1+x)(1+x^2)(1+x^3)\cdots(1+x^9)=R(x)+x^9R(x)$$  So Coefficient of $x^9$ in $P(x)$ is $1$+coeff of $x^9$ in $ R(x)$ where $$R(x)=(1+x)(1+x^2)\cdots(1+x^8)$$
But the method is becoming lengthy. Can i have any good approach
 A: HINT: The desired coefficient is the number of ways to express $9$ as a sum of distinct positive integers. Since $1+2+3+4>9$, you need only consider sums of at most three integers.
A: Take some of the terms say like 1, $x^1 , x^3 , x^5 $ then their product will be $x^{9}$ with coefficient 1, similarly for other $x^9$ terms you need terms whose products give $x^9$, i.e no of ways to express 9 as a product of integers (order does not matter). 
A: You can just try to find the different ways to decompose $9$ into different integers:
$$ 9 = a_1 + a_2 + \cdots + a_k, \, \, k\in \mathbb{N}, \, 1 \le a_1 < a_2 < \cdots < a_k$$
Since those would be the only possible ways a monome in the developed expression of $G(x)$ to have degree of $9$.
A: Just as other answers, the coefficient of $x^9$ is the number of ways to express $9$ into sum of distinct integers. 
And this remind me of Euler's Partition function and the Pentagonal number theorem.
Euler has proved that for number of ways to express an integer into sum of distinct integers, $q(n)$, there is a recurrence relation:
$$q(n)=a_k+q(n-1)+q(n-2)-q(n-5)-q(n-7)+...$$
where $q(n)=0$ if $n<0$ and $a_k$ is $(−1)^m$ if $\exists m\in{\Bbb{N}},k = 3m^2 − m $ and is $0$ otherwise.
So you may reduce $q(9)$ into $q(8)+q(7)-q(4)-q(2)$ and to reduce further to make it simplier to calculate. For instance it is easy to check value of $q(0), q(1)$, etc for small value of $n$.
For more straightforward way, you may just check the sequence A000009 in OEIS. And we get $q(9)=8$ 
Reference:
Wiki
https://en.wikipedia.org/wiki/Partition_(number_theory)
OEIS
https://oeis.org/A000009
