How to evaluate this sum 2? $\displaystyle\sum_{x+y+z=2014}xy^2z^3$ $\quad , x,y,z\in\mathbb{N}$
I think it maybe use combinatorial method.

The best solution, I think, uses generating functions.
Let $A(t) = \sum_{x=1}^\infty x t^x$, $B(t) = \sum_{y=1}^\infty y^2 t^y$, 
$C(t) = \sum_{z=1}^\infty z^3 t^z$.  Then your sum is the coefficient of 
$t^{2014}$ in $A(t) B(t) C(t)$.  Fortunately, $A(t)$, $B(t)$ and $C(t)$ can be 
evaluated in closed form.  Simplify and write using partial fractions to get
a closed form for the series coefficients.
