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Results tagged with binomial-coefficients
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user 521094
For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
5
votes
3
answers
936
views
How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx) \, dx$$
by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n( …
- 513
4
votes
Accepted
How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?
I got it ...
firstly the degree of $(x^p+1)^p$ is $p^2$ So the degree of $\prod_{p=1}^n (x^p+1)^p$ is
$$N=1+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$
now we have
$$\prod_{p=1}^n (x^p+1)^p=\sum_{p=1}^N …
- 513
4
votes
2
answers
216
views
how to prove identity for nth derivative of $(\text{arctanh}(x))^j$?
this question asked on MSE
I worked on integral problem and I got that
$$ \int_0^1 \frac{x^n}{\ln \left(\frac{1-x}{1+x} \right) } dx=-\frac{2}{(n+1)!}\sum_{j=1}^{n+1}F(n,j) \eta'(-j)$$
where $\eta(x)$ …
- 513