Questions tagged [binomial-coefficients]
For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
476 questions
6
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2
answers
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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 ...
10
votes
3
answers
1k
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Zero sum of binomial coefficients
Is there a function $p:\mathbb N\to \{ 1,-1 \} $ and a fixed $N\in \mathbb N$ such that for every $n \geq N$ we get:
$\sum _{i=0} ^{n} p(i)\binom {n}{i}=0$
?
Obviously $p(i)=(-1)^i$ works for $N=1$...
- 337
3
votes
1
answer
1k
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Yet another sum involving binomial coefficients
Let $k,p$ be positive integers. Is there a closed form for the sums
$$\sum_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$
$$\sum_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$
(...
2
votes
1
answer
367
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Bounding a hypergeometric sum
Is there a general method for finding upper bounds of hypergeometric sums?
The sum in particular I am trying to bound is $\sum_{j=1}^{k}\binom{k-1}{j-1}\binom{n-k-1}{j-1}4^{k-j}$.
This is a ...
- 491
0
votes
2
answers
1k
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Cosine of a Partial Sum
Does anyone know of a closed formula for $\displaystyle \cos\left(\sum_{n=0}^m a_n\right)$? I've seen formulas for $\displaystyle \cos\left(\sum_{n=0}^\infty a_n\right)$ and $ \displaystyle \tan\left(\...
- 591
9
votes
3
answers
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Multiplicative Convolution for Binomial Coefficients
I know Vandermonde's convolution for binomial coefficients:
$$\sum_{j=0...k} \binom{n}{j} \binom{m}{k-j} = \binom{n+m}{k}$$
Is there a similar multiplicative convolution? More precisely, is there a ...
- 93
5
votes
1
answer
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The hexagonal property of Pascal's triangle
Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that:
the product of non-adjacent vertices is constant.
the greatest common ...
- 159
23
votes
8
answers
14k
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Lower bound for sum of binomial coefficients?
Hi! I'm new here. It would be awesome if someone knows a good answer.
Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case $\...
- 263
7
votes
0
answers
1k
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Inverse of a matrix with binomial coefficients
Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...
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9
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2
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945
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Asymptotic difference between a function and its "binomial average"
(I posted this question on Math.SE a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{...
- 3,313
6
votes
1
answer
1k
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Divisibility of a binomial coefficient by $p^2$ -- current status
While skimming the book Concrete Mathematics, (edit: first edition) I came across the following problem, which is listed there as a Research Problem: (Chapter 5, Exercise 96)
Is ${2n \choose n}$ ...
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3
votes
2
answers
2k
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How to restore the original formula from a binomial-like expansion?
I encountered with a recursive formula of the following kind:
$$A(0,x)=1$$
$$A(n,x)= \sum _{j=0}^{n-1} \binom{n-1}{j} A(n-j-1,x) \sum _{k=0}^{x-1} A(j,k)$$
The sum terms can be re-arranged so to ...
- 10.4k
9
votes
3
answers
2k
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Solving a general two-term combinatorial recurrence relation
What is known about explicit (not necessarily closed-form) solutions to the recurrence
$$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$
with initial condition $R_0^0 = 1$ and ...
- 3,313
18
votes
1
answer
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How big is the sum of smallest multinomial coefficients?
Given positive integers $n$ and $d$, let $S$ indicate the list of all $d$-tuples of non-negative integers $(c_1,\ldots,c_d)$ such that $c_1+\cdots+c_d=n$. Let $v_i$ be the value of the multinomial ...
- 2,384
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votes
2
answers
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Combinatorial proof of a recurrence for the Catalan numbers
I would like to ask whether there is a combinatorial proof of the following recurrence relation for Catalan numbers:
$$
C_{n+1}=\frac{4n+2}{n+2} C_n.
$$
Thanks!~
- 459
9
votes
1
answer
2k
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Partial sums of the Chu--Vandermonde identity
I am interested in finding a lower bound of the sum:
$$\sum_{i=0}^d \left(\genfrac{}{}{0pt}{}{n}{i}\right)
\left(\genfrac{}{}{0pt}{}{m}{k-i}\right)$$
when $d < k$ (and assuming both $n\geq k$, $m\...
- 91
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votes
3
answers
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Is the sum $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0?$
I am trying to prove $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$. This inequality has been verified by computer for $k\le40$.
Some clues that might work (kindly provided by ...
1
vote
2
answers
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bound for binomial coefficients
How can one show $\displaystyle\frac{(m+n-1)!}{m ! (n-1)!}\leq \left[\frac{e (m+n-1)}{n}\right]^{n-1}$ ?
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35
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3
answers
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A binomial generalization of the FLT: Bombieri's Napkin Problem
This is an extract from Apéry's biography
(which some of the people have already enjoyed in
this answer).
During a mathematician's dinner in
Kingston, Canada, in 1979, the
conversation turned to ...
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13
votes
4
answers
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Rational congruence of binomial coefficient matrices
Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on.
Say that two matrices $A$ and $B$ over the ...
- 88.3k
48
votes
5
answers
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Integer-valued factorial ratios
This historical question recalls
Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation
Chebyshev used the factorial ratio sequence
$$
u_n=\frac{(30n)!n!}{(15n)!(10n)!(6n)...
- 13.7k
18
votes
2
answers
3k
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Binomial supercongruences: is there any reason for them?
One of the recent questions, in fact
the answer
to it, reminded me about the binomial sequence
$$
a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2,
\qquad n=0,1,2,\dots,
$$
of the Apéry ...
- 13.7k
10
votes
3
answers
2k
views
Showing a matrix is negative definite [formerly Showing a sum is always positive]
For each $d$, I have a matrix $M$ with values
$$
M_{ij} = \begin{cases}
\frac{4ij}{d} - \binom{2d}{d} & i \neq j & \\\\
\frac{4i^2}{d} - \binom{2d}{d} -
\...
- 491
128
votes
17
answers
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Sum of 'the first $k$' binomial coefficients for fixed $N$
I am interested in the function $$f(N,k)=\sum_{i=0}^{k} {N \choose i}$$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other ...
- 1,358
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5
answers
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Asymptotics of a Bernoulli-number-like function
Tony Lezard asked me the following question which seemed like it should not be too hard but which I did not immediately see how to answer. Define $f(n,k)$ recursively by $f(1,k) = 1$ and
$$f(n,k) = \...
- 88.3k
2
votes
3
answers
701
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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 ...
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