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Results tagged with recurrences
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user 521094
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