# Tagged Questions

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### Ordered partitions of an integer (with a twist)

I would like to know how to prove (preferably algebraically) that $P_1(2,n)=F_{2n+1}$, where $P_1(2,n)$ is what I define to be the number of ordered partitions of an integer, where the number $1$ has ...
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### Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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### Evaluate $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

How to find the value (if possible) of this formula? $$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$ where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.
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### How to calculate the sum of combinatorial numbers

For my work on an almost completely unrelated field I came across the following formula. I know that I have learned that all in high school. But since this is more than 15 years ago in which I never ...
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### Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
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### An equation on Catalan number [closed]

Catalan numbers have the form $C_n=\frac{1}{n+1}\binom{2n}{n}$ prove: $C_{n+1}=\sum_{m+k=n}C_mC_k$ I tried to expand $C_n$ but soon get confused..
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### Showing that two sums are equivalent

given \begin{gather} U_d(x,y,q\mid i_1,\ldots,i_k)=\sum\limits_{n,m\geq0}x^ny^m\sum\limits_{\sigma = i_1\ldots i_k\sigma_{k+1}\ldots\sigma_m\in C_{[d]}(n,m)}q^{v(\sigma)}. \end{gather} show ...
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### The sum of palindromes from 100 to 900

I'm working with palindromes from $100-999$. I'm having trouble with the step highlighted in red. Can someone explain the algebra to me? Taken from: Discrete and Combinatorial Mathematics: An ...
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### Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $${\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}}$$ what I ...
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### Sum of combination equilvalence

Could someone explain to me why the identity $$\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$$ holds?
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### What's the interpretation of $\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$?

I'm having problems with finding the combinatorial interpretation of this sum: $$\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$$ Can anyone help, please?
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### Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
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### How to prove this identity [closed]

I would like to prove the following identity without using induction: $$\sum _{ k=1 }^{ n }{ { (-1) }^{ k } {n\choose k} }\cdot k^n=(-1)^n\cdot n!.$$
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### Does $\sum_{k=0}^{k=n} {n \choose k} k!$ have a closed form for integers $k,n$?

While doing research in computer system, I came across the following summation: $$S_n = \sum_{k=0}^{n} {n \choose k} k! = \sum_{k=0}^{n} \frac{n!}{(n-k)!}$$ where both $n$ and $k$ are integers. $S_n$ ...
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### Sum of combinations with a condition

Let $m,n,p,q,r$ be non-negative integers, with $0<m\leq n$ and $p+q+r=n$ The identity $\binom{n}{m}=\sum_{x+y+z=m}\binom{p}{x}*\binom{q}{y}*\binom{r}{z}$ holds? I already checked it for m=2, n=5. ...
Can this summation be simplified ($A$ is a constant)? $$\displaystyle\sum_{i=1}^{A} \displaystyle\sum_{j>i}^{A} f(i)f(j)$$ By simplified I mean either a closed-form expression in terms of $A$ ...