In the expansion of $(1+x)(1-x^2)(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3})^5, x \neq 0$,the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to

The numerically greatest term in the expansion of $(x+3y)^{13}$,when $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is

The coefficient of $x^5$ in the expansion of $\left(2 x^3-\frac{1}{3 x^2}\right)^5$ is

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})^{n}$,in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}: 1$. If the sixth term from the beginning is $\frac{\alpha}{\sqrt[4]{3}}$,then $\alpha$ is equal to $.......$

Let $a_n$ denote the term independent of $x$ in the expansion of $\left[x+\frac{\sin(1/n)}{x^2}\right]^{3n}$. Then $\lim_{n \to \infty} \frac{a_n \cdot n!}{^{3n}P_n}$ equals

The sum of the coefficients of $x^{32}$ and $x^{-31}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$ is: