If $(1+x)^{15}=a_0+a_1 x+\ldots+a_{15} x^{15}$,then $\sum_{r=1}^{15} r \frac{a_r}{a_{r-1}}$ is equal to

The term independent of $x$ in the expansion of $(1-3x+2x^3)(\frac{3x^2}{2}-\frac{1}{3x})^9$ is

If the constant term in the binomial expansion of $\left(\frac{x^{5/2}}{2} - \frac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha}\beta$,where $\beta < 0$ is an odd number,then $|\alpha\ell - \beta|$ is equal to

Suppose $l, m, n$ 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 $a^2: b^2=$

Find the coefficient of $\alpha^t$ in the expansion of $(\alpha + p)^{m - 1} + (\alpha + p)^{m - 2} (\alpha + q) + (\alpha + p)^{m - 3} (\alpha + q)^2 + \dots + (\alpha + q)^{m - 1}$,where $\alpha \neq -q$ and $p \neq q$.

The coefficient of $x^{18}$ in the product $(1+ x)(1- x)^{10} (1+ x + x^2 )^9$ is