# Tagged Questions

**1**

vote

**0**answers

128 views

### Asymptotic behaviour of sequence

I am interested in the sequence
$$a(n)=\sum_{k=0}^n {p(n-k)-1 \choose k}$$
where $p(n)=(r-1)n^2+(2r-1)n+r$ for some $r \in \mathbb{N}$ or more generally any polynomial equation.
When $r=1$ this ...

**1**

vote

**1**answer

289 views

### Limit of an infinite series, related to (generalised Newton's) binomial expansion

How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$:
...

**6**

votes

**2**answers

669 views

### Closed form or/and asymptotics of a hypergeometric sum

Dear mathematicians,
I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed ...

**9**

votes

**3**answers

945 views

### Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence
$$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$
with initial condition $R_0^0 = 1$ and ...

**17**

votes

**3**answers

2k views

### Need help proving that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$

Hello.
I have been trying very hard to show that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$ and could not quite get anywhere. This inequality has been verified by computer ...