All Questions
Tagged with summation combinatorics
1,931
questions
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Change order of summation
I need to change the summation order in the sum:
$$
\sum\limits_{n=0}^{\infty} \sum\limits_{k=0}^{ n m } F(k, n - mk)
$$
In Srivastava's book, I came across a similar formula
\begin{array}{c}
\sum\...
- 8,194
-2
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2
answers
105
views
Reference for ${}_nC^0 + {}_nC^1 + {}_nC^2 + \cdots + {}_nC^n = 2^n$ [closed]
How can I find, or what is a good reference for:
$${}_nC^0 + {}_nC^1 + {}_nC^2 + \cdots + {}_nC^n = 2^n$$
I could write
References
[1] Binomial sums, Binomial Sums -- from Wolfram MathWorld
but I need ...
- 15
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votes
1
answer
61
views
What is $\sum_{m=0}^{\lfloor k/3\rfloor}{2k\choose k-3m}$ [closed]
With the floor function, I am not sure how to approach this.
Edit: I have a formula $\frac{1}{6}\left(4^k+2+3\frac{(2k)!}{(k!)^2} \right)$, but I got it purely from guess work.
- 101
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Closed form solution for $\displaystyle \sum _{i=r+1}^{k}\frac{1}{i-1} \cdot \frac{( n-i) !}{( k-i) !}$, where $n,k,r$ are constants and $r \leq k<n$ [closed]
Context
I arrived at this summation, while computing a probability of The Secretary Problem.
Question:
Find Closed form for this or if possible simplify
$$
\frac{r}{n}\sum_{i = r + 1}^{k}\frac{1}{i - ...
- 17
2
votes
2
answers
45
views
Given a set of integers, and the number of summations find the resulting frequencies
Given a set $X = \{x_1,x_2,...x_m\}\subset \mathbb{Z}$ and the number of allowed addends $N$. How can I find the frequency of every possible sum?
Example: $X = \{-1, 2\}$ and $N=3$ then every ...
- 389
-3
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How can I show that summing $\sum_{i=1}^n\binom{i+1}2$ equals $\binom{n+2}3 $ [closed]
How can I show that $$\sum_{i=1}^n\binom{i+1}2=\binom{n+2}3 ?$$
- 298
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0
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43
views
Closed form for nested sum involving ratios of binomial coefficients
I ran into the following nested sum of binomial coefficients in my research, but I couldn't find the closed form expression for it. I looked at various sources and still couldn't find the answer. So I ...
2
votes
0
answers
95
views
Fractional part of a sum
Define for $n\in\mathbb{N}$ $$S_n=\sum_{r=0}^{n}\binom{n}{r}^2\left(\sum_{k=1}^{n+r}\frac{1}{k^5}\right)$$
I need to find $\{S_n\}$ for $n$ large where $\{x\}$ denotes the fractional part of $x$.
$$...
- 840
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29
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The sum of the signed stirling numbers times the factorials
The sum of interest is the following...
$$
\sum^{n}_{k=1} (k-1)!s(n,k)
$$
Where the $s(n,k)$ are the signed Stirling number of the first kind. The sum is very close to other identities involving the ...
- 692
5
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0
answers
58
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Vanishing of Stirling number of second kind $S(n,k)$ for $k>n$ [duplicate]
Apologies for this question, but I feel I am missing some sort of trivial observation somehow.
Stirling numbers of the second kind are given by the series: $$S(n,k)=\frac{1}{k!} \sum_{i=0}^{k} (-1)^i {...
3
votes
1
answer
78
views
Combinatorial Proof of the $\sum_{k=1}^{n} k^2 \binom{n}{k} = n(n + 1) 2^{n - 2}$ [duplicate]
How to prove that $$\sum_{k = 1}^{n}k^2\binom{n}{k} = n(n + 1)2^{n -2}$$ in an combinatorial way?
I have a algebraic proof method, but I don't know how to use a combinatorial proof method to do it.
- 31
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votes
2
answers
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high school math: summands
Let's say we have a question that asks you to find the amount of all possible integers adding up to a random number, lets just say 1287. However, the possible integers is restricted to explicitly 1's ...
- 31
1
vote
1
answer
105
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Spivak Exercise, Prove Vandermonde's Identity $\sum_{k=0}^{l}\binom{n}{k}\binom{m}{l-k}=\binom{n+m}{l}$
Prove that
$$\sum_{k=0}^{l}\binom{n}{k}\binom{m}{l-k}=\binom{n+m}{l}$$
Hint: Apply the binomial theorem to $(1+x)^n(1+x)^m$
Proof:
Following Spivak's advice, we have
$\sum_{k=0}^{n}\binom{n}{k}x^k=(...
- 115
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0
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44
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Choosing an ordered triplet of non-negative integers $(m_1, m_2, m_3)$ such that $m_1 + m_2 + m_3 = n$.
Define
$$A = \{ (m_1, m_2, m_3) : m_1 \geq 0, m_2 \geq 0, m_3 \geq 0, m_1 + m_2 + m_3 = n\}$$
Given that $n \geq 0$ and $n, m_1, m_2, m_3 \in \mathbb{Z}^+$, then why it is the case that
$$\vert A \...
- 934
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2
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44
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Trying to prove equivalence of combinatorial formula and nested summations
I’m sorry if this is a dumb problem, but I’m trying to get into mathematics and prove this but I’m only in 9th grade and haven’t found any sources on this:
Prove that $$ {n \choose r} = \overbrace{ \...
