I got it partially as finding coefficient $x^{n}$ using Newton binomial expansion is not a human task, computers should be used.
The expansion looks like this:-
$x_{1}\in {(0,3,6,9,12.....)} \\ x_{2}\in {(0,4,8,12,16....)} \\ x_{3}\in {(0,2,4,6,8,.....)} \\ x_{4}\in {(0,5,10,15,20,.....)}$
$\begin{align*} &= [x^{n}](1+x^{3}+x^{6}........)(1+x^{4}+x^{8}+....)(1+x^{2}+x^{4}+....)(1+x^{5}+x^{10}+....)\\ &=[x^{n}]\frac{1}{1-x^{3}}.\frac{1}{1-x^{4}}.\frac{1}{1-x^{2}}.\frac{1}{1-x^{5}} \\ &=[x^{n}]\frac{1}{(1+x)^{2}}.\frac{1}{(1-x)^{2}}.\frac{1}{1+x^{2}}.\frac{1}{1-x^{3}}.\frac{1}{1-x^{5}} \\ &= [x^{n}]? \end{align*}$