For $a > 0$,let $\frac{1}{a(a+1)(a+2) \ldots(a+20)}=\sum_{k=0}^{20} \frac{A_{k}}{a+k}$. Then the value of $100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^{2}$ is equal to $....$

If $\frac{x^2-2}{(x^2+1)(x^2+3)} = \frac{Ax+B}{x^2+1} + \frac{Cx+D}{x^2+3}$,then $D=$

For $|x| < \frac{1}{2}$,if the coefficient of $x^{10}$ and the constant term in the expansion of $\frac{2 x^3+8 x^2-2 x-2}{(1-x)(1+x)(1-2 x)}$ in powers of $x$ are $l$ and $m$ respectively,then $lm=$

The absolute value of the difference of the coefficients of $x^4$ and $x^6$ in the expansion of $\frac{2 x^2}{(x^2+1)(x^2+2)}$ is

If $\frac{3x}{(x-a)(x-b)} = \frac{2}{x-a} + \frac{1}{x-b}$,then $a:b$ is equal to

If $\frac{3 x^4-2 x^2+1}{(x-2)^4}=A+\frac{B}{x-2}+\frac{C}{(x-2)^2}+\frac{D}{(x-2)^3}+\frac{E}{(x-2)^4}$,then $2 A+3 B-C-D+E=$