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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 ...
Faoler's user avatar
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2 votes

How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?

As suggested by @StevenStadnicki above, I'll formulate my comment as an answer. For $n\to\infty$, the coefficients of $$ \tag{1}\label{eq:1} P_\infty(x) = \lim_{n\to\infty}P_n(x) = \prod_{k=1}^\infty ...
Fred Hucht's user avatar
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How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?

Caveat: OP asked me in the comment section how he can calculate the coefficient explicitly. This answer is mainly algorithmic (dynamic programming) and straight forward with FFT/convolutions/dynamic ...
AspiringMat's user avatar
7 votes
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An integer sequence related to Pascal’s triangle

Yes, it is related to known sequences. Polynomials $(k+1)!n_k(p)$, where $p$ is treated as a variable, have coefficients triangle $$ 1,\\ -3,1,\\ 20,-9,1,\\ -210,107,-18,1,\\ ................... $$ ...
Cave Johnson's user avatar
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7 votes
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About the exact origin of a binomial congruence

Dickson in his "History of the theory of numbers. Vol. I: Divisibility and primality." (see page 64) attributes this statement to Genty (Histoire et mem. de I'acad. roy. sc. insc. de ...
Alexey Ustinov's user avatar

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