All Questions
154
questions
2
votes
2
answers
85
views
Limit of $\sum_{j=0}^n (1-j){n\choose j} \frac{1}{(n-1)^j}\left( \frac{n-j}{n} \right)^k$ as $n\to \infty$ (and $k=n$)
Question
Find $\lim_{n\to \infty} P(n,n)$ where
$$
P(n,k) = \sum_{j=0}^n (1-j){n\choose j} \frac{1}{(n-1)^j}\left( \frac{n-j}{n} \right)^k.
$$
The origin of this sum is from this question. The ...
- 9,748
2
votes
4
answers
155
views
A Combinatoric Identity Involving Binomial Coefficients: $\sum^{n}_{r=1} {n \choose r} (-1)^r \frac{2^r-1}{2r}= \sum^{n}_{r=1}\frac{(-1)^r}{2r}$
Recently I came across this combinatoric identity:
$$
\begin{equation}
\sum^{n}_{r=1} {n \choose r} (-1)^r \frac{2^r-1}{2r}
\end{equation} = \sum^{n}_{r=1}\frac{(-1)^r}{2r}
$$
I have verified that ...
- 23
3
votes
0
answers
468
views
Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing
Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties:
a) $na_n \in \mathbb{N} \setminus \{0\}$ for every $n \in ...
- 637
1
vote
1
answer
129
views
Prove $\sum_{k=j}^{\lfloor n/2\rfloor}\frac1{4^k}\binom{n}{2k}\binom{k}{j}\binom{2k}{k}=\frac1{2^n}\binom{2n-2j}{n-j}\binom{n-j}j$
let $x>1$, $n\in \mathbb N$ and
$$P_{n}(x)=\dfrac{1}{\pi}\int_{0}^{\pi}(x+\sqrt{x^2-1}\cos{t})^ndt.$$
Prove that
$$P_{n}(x)=\dfrac{1}{\pi}\int_{0}^{\pi}\dfrac{1}{\left(x-\sqrt{x^2-1}\cos{t}\right)^{...
- 137
0
votes
1
answer
69
views
Infinite Sum with Geometric and Combinatorial terms
I was working on a probability question which at one point brought me to the sum:
$$\sum_{k=0}^{\infty} \Big{(}\frac{4}{6}\Big{)}^k {4+k \choose k}$$
I put it in Wolfram Alpha and obtained the answer ...
3
votes
1
answer
96
views
Sum of factorials / binomial coefficients
I came across this identity:
$$\sum_{k=0}^n \frac{n!(n+1)!}{(n-k)!(n+k+1)!}(2k+1) = n+1. $$
This shows up quite undirectly when considering spherical harmonics, I would be interested in a direct ...
- 187
1
vote
0
answers
34
views
Sum : $\sum_{t=1}^{r} t\binom{n-t}{n-r}=\binom{n+1}{r-1}$ [duplicate]
We want to prove that:
$$\sum_{t=1}^{r} t\binom{n-t}{n-r}=\binom{n+1}{r-1}$$
I have used basic properties of binomial coefficients like $$\binom{n+1}{r+1}= \frac{n+1}{r+1}\binom{n}{r}$$ but they ...
- 2,821
0
votes
1
answer
64
views
Simplifying a series with multiset coefficients
I am struggling to simplify this series:
$$\sum_{k = 0}^{\infty} \left(\!\!{n\choose k}\!\!\right) a^k k^2 =
\sum_{k = 0}^{\infty} {n + k - 1\choose k}\ a^k k^2.$$
I understand that the square under ...
2
votes
1
answer
100
views
For what $a \in N$ , Is $\sum_{k=0} ^ n (-1)^k k^a \binom nk = 0 ?$ [duplicate]
Background:
Let $n>2$ and $ n \in N$.
$$\sum_{k = 0}^n (-1)^k \binom nk (x-k)(y-k)(z-k)= 0$$ (not verbatim)
I was solving the above textbook problem. As a part of my solution, I came across the ...
- 2,821
-1
votes
1
answer
135
views
How to efficiently compute $\sum_{k} \prod_{i=1}^N \binom{N-k_i}{k_i} \frac{N}{N-k_i}$
I want to evaluate the sum
$$
\sum_{k} \prod_{i=1}^N \binom{N-k_i}{k_i} \frac{N}{N-k_i}
$$
where the sum ranges over all ordered partitions $(k_1, \dots, k_N)$ such that $\displaystyle\sum_{i=1}^N ...
- 470
0
votes
1
answer
387
views
How to find the sum of C(n;k) for odd k? [duplicate]
How to calculate this expression?
$$\sum_{k \text{ odd}}{n\choose k}$$
I only just notice that, if n odd,then $$\sum_{k \text{ odd}}{n\choose k} = 2^{(n-1)}$$
- 37
0
votes
1
answer
64
views
How to find the number of ways to split a array into 2 equal subsets?
I need to find the number of ways to split the array into 2 equal subsets (not necessarily all elements must be in at least one of the subsets). The array contains numbers from $1$ to $N$. For example:...
- 37
17
votes
4
answers
647
views
Bernoulli numbers identity:$\sum_{k=0}^n\sum_{l=0}^m\binom{n}{k}\binom{m}{l}\frac{(n-k)!(m-l)!}{(n+m-k-l+1)!}(-1)^l B_{k+l}=0$,for all $n\ge1$,$m\ge0$
For all $n\geq 1$ and $m\geq0$, I'm trying to prove that
$\sum_{k=0}^n\sum_{l=0}^m\binom{n}{k}\binom{m}{l}\frac{(n-k)!(m-l)!}{(n+m-k-l+1)!}(-1)^l B_{k+l}=0$
where $B_n$ are the Bernoulli numbers with $...
- 181
0
votes
1
answer
146
views
How to find $\sum_{k=1}^n k\binom{n}{k}?$ [duplicate]
How to find the sum of series,
$$\sum_{k=1}^n k\binom{n}{k}?$$
I try solve using this fact
$$\sum_{k=1}^n \binom{n}{k} = 2^n$$
but it didn't help.
And now i haven't some ideas.
Maybe i can calculate $$...
- 37
5
votes
2
answers
178
views
Prove that $\sum_{k = r}^{n} k(k - 1)(k - 2)\cdots(k - r+ 1)D_n(k) = n!$
Prove that $$\sum_{k = r}^{n} k(k - 1)(k - 2)\cdots(k - r+ 1)D_n(k) = n!$$ where $D_n(k)$ represents the number of permutations of length $n$ with exactly $k$ fixed points.
Hello, I am trying this ...
- 1,858