suppose $n$ , $k$ are positive integers both. if for any set $ \{ a_1,a_2,....,a_k \}$ with sum of $n$ we write down multiplication of $a_1,a_2,....,a_k$. now I want to prove that the sum of written numbers is $\binom{n+k-1}{2k-1}$ .
I am trying to solve a problem from Stanley's book, it says:  Fix $k,n \in \mathbb{P}$. Show that: \begin{align} \sum a_1 a_2 \cdots a_k = \binom{n+k-1}{2k-1} \end{align} where the sum ranges over all compositions $(a_1 , a_2 , \ldots , a_k)$ of $n$ into $k$ parts.  I am trying to reason like this:  we need to find the coefficient $c_n = \sum a_1 a_2 \cdots a_k$ from this generating function --  \begin{align} \sum_n c_n x^n &amp;= \sum_n \sum a_1 a_2 \cdots a_k x^n \\  &amp;= \sum_n \sum a_1 a_2 \cdots a_k x^{a_1 + a_2 + \cdots + a_k}\\ &amp;= \sum_n \sum a_1x^{a_1} a_2x^{a_2} \cdots a_kx^{a_k} \end{align}  after that, I have no clue, how do I solve this ?   moreover, what is the range in the inner sum ?    If we consider Mark Riedel's answer, and assume $n=4$, $k=2$; then the sum will be  \begin{align} \sum (z + 2z^2)^2 = z^2 + 4z^3 + 4z^4 \end{align}  On the other hand the compositions will be $(1,3), (2,2), (3,1)$, therefore the above sum will be counted as:  \begin{align} (1.3)z^{1+3} + (2.2)z^{2+2} + (3.1)z^{3+1} &amp;= 1z^1.3z^3 + 2z^2.2z^2 + 3z^3.1z^1\\  &amp;= 3z^4 + 4z^4 + 3z^4 = 10z^4 \end{align}   what's going on? what am I missing?
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Evaluating $\int \frac{1}{{x^4+1}} dx$ I am trying to evaluate the integral  $$\int \frac{1}{1+x^4} \mathrm dx.$$  The integrand $\frac{1}{1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I can find the partial fraction of $\frac{1}{1+x^4}$. But I failed to factorize $1+x^4$.   Any other methods are also wellcome. examples:         Gift (German) = poison   poisson (French) = fish   embarazada (Spanish) = pregnant   sauce (Spanish) = willow   triviale (Italian) = vulgar   parentes (Portuguese) = relatives   slim (Dutch) = smart Execution order of event receivers and workflow on a list