Divisibility of an Evenly Spaced Binomial Coefficient Series

Question

By this paper, it can be shown that for $n>0$ and $N\in\mathbb{N}$ $$\sum_{k=0}^n\binom{Nn}{Nk}=\frac{2^{Nn}}{N}\sum_{k=0}^{N-1}(-1)^{kn}\cos^{Nn}\left(\frac{k\pi}{N}\right)$$

Now, for a recursive sequence defined here, $$A_n(N)=-\sum_{k=0}^{n-1}\binom{Nn}{Nk}A_k(N); A_0(N)=1$$

and so

$$A_n(N)+1=-\sum_{k=1}^{n-1}\binom{Nn}{Nk}A_k(N)$$

From here it can be easily obtained that $A_1(N)=-1$. Now I originally wanted to show that for certain values, we have that

$$A_n(N)+1\equiv 0\pmod{N+1}$$

But this seems to be only true for certain values of $N$, where $N=2,3,4,6$. I determined this by an induction argument, whose base case is above. For all $0<k<m$, we assume $A_m(N)+1\equiv 0\pmod{N+1}$. This means that $A_m(N)\equiv N\equiv -1\pmod{N+1}$ and thus $A_m(N)=(N+1)q-1$ for $q\in\mathbb{Z}$. Substituting into the above recursion yields

$$A_m(N)+1=-\sum_{k=1}^{m-1}\binom{Nm}{Nk}[(N+1)q-1]=-(N+1)\sum_{k=1}^{m-1}\binom{Nm}{Nk}q+\sum_{k=1}^{m-1}\binom{Nm}{Nk}$$

The first RHS sum is necessarily divisible by $N+1$, so the second sum in question would have to be divisible by $N+1$. However this is not always the case. To see its divisibility by $2,3,4$ and $6$, note that

\begin{eqnarray*}\sum_{k=1}^{m-1}\binom{Nm}{Nk}&=&\sum_{k=0}^{m}\binom{Nm}{Nk}-\binom{Nm}{0}-\binom{Nm}{Nm}\\&=&\frac{2^{Nm}}{N}\sum_{k=0}^{N-1}(-1)^{km}\cos^{Nm}\left(\frac{k\pi}{N}\right)-2\end{eqnarray*}

And from this form, we can show the validity of divisibilities for $N=2,3,4,6$ by cases (replacing $N$ and working out the residue under modulus $(N+1)$)

My question is this. I KNOW that it is not true for all $N$ through a proof presented here. Is there a way to either

1) Find more values of $N$ where it holds

2) if not, prove that for the remaining values of $N$, the hypothesis is wrong?

The values of $N$ chosen $(2,3,4,6)$ are based due to the fact the formula above involves cosines and take on simple values to calculate algebraically. I don't know how to show that it is invalid for the remaining or how to find other values of $N$. Can anyone help guide the way?

EDIT: So I used Mathematica to do a calculation of the divisibility up to $N=50$. Using the binomial sum instead of the cosine sum, I was able to see that for the first 50 numbers of each recursion for $N$, the values of $N$ that seem to produce the divisibility results are those numbers that are prime powers minus 1, $p^\alpha$, for prime $p$ and positive integer $\alpha$. This list is found as A181062. Can anyone confirm this?

Answer 1

An interesting special case is when $N+1$ is prime. In that case we prove that $\binom{Nm}{Nk}$ is divided by $N+1$ when $m\le N+1$. By expanding the terms we obtain$$\binom{Nm}{Nk}=\dfrac{Nm(Nm-1)\cdots (Nk+1)}{(Nm-Nk)(Nm-Nk-1)\cdots 2\cdot 1}$$If $N+1$ is prime then the only numbers $x$ such that $\gcd(N+1,x)>1$ are the integer multiples of it i.e. $$x=(N+1)\cdot l\qquad,\qquad l=1,2,3,\cdots$$notice that if $m\le N+1$ there are $m-k$ multiples of $N+1$ among $Nm,Nm-1,\cdots Nk+1$ and $m-k-1$ such multiples among $Nm-Nk,Nm-Nk-1,Nm-Nk-2,\cdots 2,1$ (this is a simple counting) therefore $$\binom{Nm}{Nk}=\dfrac{Nm(Nm-1)\cdots (Nk+1)}{(Nm-Nk)(Nm-Nk-1)\cdots 2\cdot 1}=\dfrac{a(N(m-1)+m-1)(N(m-2)+m-2)\cdots (Nk+k)}{b(N(m-k-1)+m-k-1)\cdots (N+1)N\cdots 2\cdot 1}=\dfrac{a'}{b'}(N+1)$$this is because after cancelling the multiples of $N+1$ from both numerator and denominator there still exists one $N+1$ which can't be cancelled out any further and both $a'$ and $b'$ are coprime with $N+1$. Since $\dfrac{a'}{b'}(N+1)$ is an integer, it turns out that $b'|a'$ since $\gcd(b',N+1)=1$ by Euclid's lemma. This completes the proof on $$N+1|\binom{Nm}{Nk}\qquad,\qquad {N+1\text{ is a prime}\\ m\le N+1\\1\le k\le m-1}$$by a simple substitution this means that$$N+1|A_m(N)+1\qquad,\qquad {N+1\text{ is a prime}\\ m\le N+1\\1\le k\le m-1}$$P.S. finding all such $N$s is really hard so i think we need to bear with special and sufficient cases.

Answer 2

The following should have been a comment but i do not have enough points for a comment. I am just giving a different perspective here..The answer is not given..

Let $N=p^{a}$ for some prime $p$ and positive integer $a$. Let $p > n$. Then \begin{equation} (1+x)^{Nn} \mod p = \sum_{k=0}^{n} {Nn \choose Nk} x^{Nk} \mod p \end{equation}