<s>[INST] Find the coefficient of $ x^6 $in the given expansion. Find the coefficient of $x^6$ in $\left[(1+x)(1+x^2)^2(1+x^3)^3 \cdots (1+x^n)^n\right]$.

Expansion
$$\left[\left(1+\binom11x \right)\left(1+\binom21x + \binom22x^2 \right)\left(1 +\binom31x+\binom32x^2+\binom33x^3 \right)\cdots \left(1 + \binom n1 x + \binom n2x^2 + \cdots +\binom n nx^n \right) \right]$$
I expanded it as shown above but couldn't proceed further. Any Help would be appreciated.  
 [/INST] Here we give a supplement regarding the cases $1\leq n\leq 5$. It is convenient to use the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$ in a series.

We obtain
Cases $n=1,2$:
\begin{align*}
  [x^6](1+x)=[x^6](1+x)(1+x^2)^2\color{blue}{=0}
\end{align*}
Case $n=3$: 
\begin{align*}
  \color{blue}{[x^6]}&\color{blue}{(1+x^3)^3(1+x^2)^2(1+x)}\\
  &=[x^6](1+3x^3+3x^6)(1+2x^2+x^4)(1+x)\tag{1}\\
  &=\left([x^6]+3[x^3]+3[x^0]\right)(1+2x^2+x^4)(1+x)\tag{2}\\
  &=0+3\cdot2+3\cdot 1\\
  &\,\,\color{blue}{=9}
\end{align*}
Case $n=4$: 
\begin{align*}
  \color{blue}{[x^6]}&\color{blue}{(1+x^4)^4(1+x^3)^3(1+x^2)^2(1+x)}\\
  &=[x^6](1+4x^4)(1+3x^3+3x^6)(1+2x^2+x^4)(1+x)\\
  &=[x^6](1+3x^3+4x^4+3x^6)(1+2x^2+x^4)(1+x)\tag{3}\\
  &=\left([x^6]+3[x^3]+4[x^2]+3[x^0]\right)(1+2x^2+x^4)(1+x)\\
  &=0 + 3\cdot 2+4\cdot 2+3\cdot 1\\
  &\,\,\color{blue}{=17}
\end{align*}
Case $n=5$:
\begin{align*}
  \color{blue}{[x^6]}&\color{blue}{(1+x^5)^5(1+x^4)^4(1+x^3)^3(1+x^2)^2(1+x)}\\
  &=[x^6](1+5x^5)(1+4x^4)(1+3x^3+3x^6)(1+2x^2+x^4)(1+x)\\
  &=[x^6](1+4x^4+5x^5)(1+2x^2+3x^3+x^4+6x^5+3x^6)(1+x)\tag{4}\\
  &=\left([x^6]+4[x^2]+5[x]\right)(1+2x^2+3x^3+x^4+6x^5+3x^6)(1+x)\\
  &=(6+3)+4\cdot 2+5\cdot 1\\
  &\,\,\color{blue}{=22}
\end{align*}

Comment:


*

*In (1) we expand the terms skipping summands with powers of $x$ greater than $6$.

*In (2) we use the rule $[x^{p-q}]A(x)=[x^p]x^qA(x)$.

*In (3) we multiply out the left-most two terms ignoring summands which do not contribute to $[x^6]$.

*In (4) we multiply out the left-most two terms and the next two terms ignoring summands which do not contribute to $[x^6]$.
</s>