TITLE: Solve a summation
QUESTION [8 upvotes]: Hi guys I have an exercise I don't know how to approach, would be cool if you could give me a tip or two! A sequence $a_{n}$ is defined by a dependency : $$ \sum_{i, j, k \geq 0}^{i+j+k = n } a_ia_ja_k = (n+1)^{\overline{5}} \text {    
      for  $n\geq$ 0}$$
I am to solve the given summation and find an equation for $a_{n}$.
Thanks in advance!

REPLY [5 votes]: Clearly
$$\sum_{n\ge 0}(n+1)^{\overline 5}x^n=D^5\sum_{n\ge 0}x^n=D^5\left(\frac1{1-x}\right)=\frac{5!}{(1-x)^6}\;.$$
As vonbrand already pointed out,
$$\sum_{i, j, k \geq 0}^{i+j+k = n } a_ia_ja_k$$
is the coefficient of $x^n$ in
$$\left(\sum_{k\ge 0}a_kx^k\right)^3\;,$$
so
$$\left(\sum_{k\ge 0}a_kx^k\right)^3=\frac{5!}{(1-x)^6}\;,$$
and
$$\sum_{k\ge 0}a_kx^k=\frac{5!^{1/3}}{(1-x)^2}=2\sqrt[3]{15}\sum_{k\ge 0}(k+1)x^k\;.$$