The coefficient of $x^{101}$ in the expression $(5+x)^{500} + x(5+x)^{499} + x^{2}(5+x)^{498} + \ldots + x^{500}$ for $x > 0$ is:

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5+(x-\sqrt{x^3-1})^5, x>1$. If $u$ and $v$ satisfy the equations $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$,then $u+v$ equals:

The coefficient of $x^{50}$ in $(1+x)^{101} (1-x+x^2)^{100}$ is......

In the expansion of $(1 + x + x^3 + x^4)^{10}$,the coefficient of $x^4$ is

Expand the expression $\left(x+\frac{1}{x}\right)^{6}$.

If $\frac{(1-px)^{-1}}{(1-qx)}=a_0+a_1x+a_2x^2+a_3x^3+\ldots$,then $a_n=$