# Tagged Questions

**1**

vote

**0**answers

150 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

**3**answers

320 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

**0**answers

68 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

**1**answer

146 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

**2**answers

442 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

**3**answers

497 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 ...