How to find determinant of this matrix? Is there a manual method to find  $\det\left(XY^{-1}\right)$ ?
Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 
1 & 3 & 3^2 & \cdots & 3^{2012} \\
1 & 4 & 4^2 & \cdots & 4^{2012} \\  
\vdots  & \vdots  & \vdots  & \cdots & \vdots  \\ 
1 & 2014 & 2014^2 & \cdots & 2014^{2012} \\ \end{array} } \right], $$
$$Y=\left[ {\begin{array}{cc}\frac{2^2}{4} & \frac{3^2}{5} & \dfrac{4^2}{6} & \cdots & \dfrac{2014^2}{2016} \\ 
2 & 3 & 4 & \cdots & 2014 \\
2^2 & 3^2 & 4^2 & \cdots & 2014^{2} \\  
\vdots  & \vdots  & \vdots  & \cdots & \vdots  \\ 
2^{2012} & 3^{2012} & 4^{2012} & \cdots & 2014^{2012} \\ \end{array} } \right] $$.
Thanks in advance.

Consider something a bit more general.  Let $$X=\left[ {\begin{array}{cc} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ 
1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\
1 & x_3 & x_3^2 & \cdots & x_3^{n-1} \\  
\vdots  & \vdots  & \vdots  & \cdots & \vdots  \\ 
1 & x_n & x_n^2 & \cdots & x_n^{n-1} \\ \end{array} } \right], $$
and
$$Y=\left[ {\begin{array}{cc}\frac{x_1^2}{x_1+r} & \frac{x_2^2}{x_2+r} & \dfrac{x_3^2}{x_3+r} & \cdots & \dfrac{x_n^2}{x_n+r} \\ 
x_1 & x_2 & x_3 & \cdots & x_n \\
x_1^2 & x_2^2 & x_3^2 & \cdots & x_n^2 \\  
\vdots  & \vdots  & \vdots  & \cdots & \vdots  \\ 
x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \cdots & x_n^{n-1} \\ \end{array} } \right]. $$
Your problem corresponds to $x_i=i+1,$ $n=2013,$ $r=2.$
Let $Y'$ be the matrix obtained by rescaling the columns of $Y$ by multiplying column $i$ by $(x_i+r)/x_i.$  So
$$\det Y'=\det Y\prod_{i=1}^n\frac{x_i+r}{x_i}
$$
and
$$Y'=\left[ {\begin{array}{cc}x_1 & x_2 & x_3 & \cdots & x_n \\ 
x_1+r & x_2+r & x_3+r & \cdots & x_n+r \\
x_1(x_1+r) & x_2(x_2+r) & x_3(x_3+r) & \cdots & x_n(x_n+r) \\  
\vdots  & \vdots  & \vdots  & \cdots & \vdots  \\ 
x_1^{n-2}(x_1+r) & x_2^{n-2}(x_2+r) & x_3^{n-2}(x_3+r) & \cdots & x_n^{n-2}(x_n+r) \\ \end{array} } \right]. $$
Now show that
$$\det Y'=-r\det X,$$
from which your determinant can easily be evaluated.  This is done by a series of row operations on $Y'.$  First subtract row $1$ from row $2.$  Then swap the first two rows.  Then divide row $1$ by $r$.  (These steps account for the factor $-r.$)  We now have the matrix
$$\left[ {\begin{array}{cc}1 & 1 & 1 & \cdots & 1 \\ 
x_1 & x_2 & x_3 & \cdots & x_n \\
x_1(x_1+r) & x_2(x_2+r) & x_3(x_3+r) & \cdots & x_n(x_n+r) \\  
\vdots  & \vdots  & \vdots  & \cdots & \vdots  \\ 
x_1^{n-2}(x_1+r) & x_2^{n-2}(x_2+r) & x_3^{n-2}(x_3+r) & \cdots & x_n^{n-2}(x_n+r) \\ \end{array} } \right]. $$
Now subtract $r$ times row $2$ from row $3$.  Then subtract $r$ times row $3$ from row $4.$  Continue in this way, finally subtracting $r$ times row $n-1$ from row $n.$  The resulting matrix will be $X^T.$
