1
vote
0answers
158 views

An extrasensory perception strategy :-)

I asked this question at MSE some months ago but I received only partial answers, so I put it here. The following sounds nice for me and I spent a good time during the investigation. But I am a ...
3
votes
3answers
344 views

Estimating a sum involving binomial coefficients [refined]

Having some work done, here is a refined version of my initial question. For integer $m>0$ and $0\le q\le m$, consider the sum $$ S(m,q) = \sum_{i=0}^{m-q} \binom{m}{i} \binom{m-i}{q}^2. $$ I ...
0
votes
0answers
72 views

Congruences for generalized Franel numbers

Let us define generalized Franel numbers $f^{(m)}_n$ through recurrence relations: $f^{(1)}_n=1$ for all $n$, and $$f^{(m)}_n=\sum\limits_{k=0}^n\binom{n}{k}^3f^{(m-1)}_k.$$ In fact ...
2
votes
1answer
155 views

Congruence for the Apery Numbers

Is it true that $$A_n\equiv (-1)^n\;\;(\mathrm{mod}\;3)\;\;?$$ Here $A_n$ is the Apery number: $$A_n=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$ What is known about congruence properties ...
6
votes
2answers
452 views

Why are negative sets multisets? (Reference request)

It is easy to establish that $$ \left({n\choose k}\right)=(-1)^k{-n \choose k}, $$ where the symbol on the left-hand-side counts the number of multisets of $k$ elements from $n$. On the Wikipedia ...
2
votes
3answers
502 views

Variant of binomial coefficients

I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc. First recall the following. If z is a ...