If sum of the coefficients of $x^r(r=0,1,2, \ldots, 2 n)$ in the expansion of $\left(1+3 x-2 x^2\right)^n$ is 128 , then $\sum_{r=1}^{2 n} r \frac{(2 n)_{C_r}}{(2 n)_{C_{r-1}}}=$

The approximate value of $\left(3 \sqrt{126}+\sin 61^{\circ}\right)$ correct to three decimal places, obtained by taking $1^{\circ}=0.0174$ radians, is

Suppose $1, m, n$ respectively represent the coefficient of $x^{10}$, the constant term and the coefficient of $x^{-10}$ in the expansion of $\left(a x^2+\frac{b}{x^3}\right)^{15}$. If $\frac{l}{m}+\frac{m}{n}=\frac{26}{11}$, then $a^2: b^2=$

For $z \in \mathbf{C}$, if $(1+z)^n=1+{ }^n C_1 z+{ }^n C_2 z^2+\ldots{ }^n C_n z^n$ and $\sum_{r=0}^{100} 100 c_r(\sin r x)=\left(2 \cos \frac{x}{2}\right)^{100} \sin k x$, then $k=$