Suppose $n$ be a given positive integer. Then the Diophantine equation $x=n$ has only $1$ solution. Just by inspection, I found that the Diophantine equation $x+2y=n$ has $\left\lfloor \dfrac{n}{2}+1\right\rfloor$ non-negative solutions for $(x,y).$
Also, according to this post the Diophantine equation $x+2y+3z=n$ has $\left\lfloor \dfrac{n^2}{12}+\dfrac{n}{2}+1 \right\rfloor$ non-negative solutions for $(x,y).$
Is there any closed form for the number of non-negative integer solutions for $(x_1,x_2,\cdots,x_k)$ of $$x_1+2x_2+3x_3+\cdots+kx_k=n$$ for a given $k\in\Bbb{N}$?
How can I prove these formulas rigorously?
EDIT
After a very tedious calculation I found that the equation $w+2x+3y+4z=n$ has $\left\lfloor \dfrac{n^3}{144}+\dfrac{5n^2}{48}+\dfrac{(15+(-1)^n)n}{32}+1 \right\rfloor$ solutions.
This solution completely agree with the approximation given by Rus May.
However still I believe that we can do something more than this.