Let $C_{r}$ denote the coefficient of $x^{r}$ in the binomial expansion of $(1+x)^{n}$, $n \in N$, $0 \leq r \leq n$. If $P_{n} = C_{0} - C_{1} + \frac{2^{2}}{3}C_{2} - \frac{2^{3}}{4}C_{3} + \dots + \frac{(-2)^{n}}{n+1}C_{n}$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.

Let $(1 + x)(1 + x + x^2)(1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{30}) = a_0 + a_1x + a_2x^2 + \dots + a_{465}x^{465}$. Then the sum $a_0 + a_2 + a_4 + \dots$ is equal to:

Let $(7 + 4\sqrt{3})^n = p + \beta$,where $n$ and $p$ are positive integers and $\beta \in (0, 1)$. Then $(1 - \beta)(p + \beta)$ is

Let $C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{10}$. If $\alpha, \beta \in R$,and $C_{1}+3 \cdot 2 C_{2}+5 \cdot 3 C_{3}+\ldots$ (up to $10$ terms) $= \frac{\alpha \times 2^{11}}{2^{\beta}-1} \left( C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots \right.$ (up to $10$ terms) $)$,then the value of $\alpha+\beta$ is equal to:

The coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

The coefficient of $x^{15}$ in the product $(1-x)(1-2x)(1-2^2x)(1-2^3x) \ldots (1-2^{15}x)$ is