## New answers tagged binomial-coefficients

3
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 ...

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 ...

2
votes

### 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 ...

7
votes

Accepted

### An integer sequence related to Pascalâ€™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
$$
1,\\
-3,1,\\
20,-9,1,\\
-210,107,-18,1,\\
...................
$$
...

7
votes

Accepted

### 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 ...

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