<s>[INST] Ramanujan's partial fraction decomposition of $\frac{1}{(x^2+a^2)\cdots(x^2+(a+n)^2)}$. \begin{align*}
\frac{1}{(x^2+a^2)\cdots(x^2+(a+n)^2)} &= \frac{2\Gamma(2a)}{\Gamma(n)\Gamma(2a+n)}\left(\frac{a}{x^2+a^2}-\frac{2a}{1!}\frac{n-1}{n+2a}\frac{a+1}{x^2+(a+1)^2}\right. \\
& \qquad + \left.\frac{2a(2a+1)}{2!}\frac{(n-1)(n-2)}{(n+2a)(n+2a+1)}\frac{a+2}{x^2+(a+2)^2}-\cdots\right).
\end{align*}
The preceding was by Ramanujan, appearing in one of his notebooks. How does one prove this? 
Especially interesting is motiving the proof:  given only the complete fraction on the left,  is there a method that makes the right side almost immediately obvious? (Basically, it would be nice if the answers imagined the RHS didn't exist in the above equation).
 [/INST] First of all there is a error in the expression on the left hand side. The correct right hand side is:
\begin{equation}
lhs=\frac{1}{(x^2+a^2)\dot \cdots \dot (x^2+(a+n-1)^2)}
\end{equation}
The left hand side is correct.
Now from partial fraction decomposition the right hand side clearly equals:
\begin{equation}
rhs = \sum\limits_{k=0}^{n-1} \frac{1}{x^2+(a+k)^2} \cdot C_k 
\end{equation}
where:
\begin{eqnarray}
C_k&=&\frac{1}{\prod\limits_{j=0,j\neq k}^{n-1} (-(a+k)^2 + (a+j)^2)}=\frac{1}{\prod\limits_{j=0,j\neq k}^{n-1} (2 a+k+j)(j-k)} \\
&=&\frac{(-1)^{k}}{(2a+k)^{(k)} (k)! (2a+2 k+1)^{(n-1-k)}(n-1-k)!}\\
&=& \frac{1}{\Gamma(n)} \cdot \binom{n-1}{k} \cdot (-1)^k \cdot \frac{\Gamma(2a+k)}{\Gamma(2 a+n+k)} (2a+2 k) \\
&=& \frac{2 \Gamma(2a)}{\Gamma(n) \Gamma(2 a+n)} \cdot \binom{n-1}{k} (-1)^k \cdot \frac{(2 a)^{(k)}}{(2a+n)^{(k)}} \cdot (a+k)
\end{eqnarray}
which is exactly what we have on the right hand side. Here of course $k=0,\cdots,n-1$.
</s>