All Questions
Tagged with binomial-coefficients polynomials
188
questions
1
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Asymtotic of some binomial sum
Assume $n$ is a positive odd integer, I need to find the asymptotic as $n$ goes to infinity of the sum
$$s(n,x)=\frac1x\sum_{k=0}^n (-1)^k\binom{-x-\frac12}{k}\binom{x-\frac12}{n-k},$$
where the ...
- 134
3
votes
1
answer
162
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Prove combination of polynomials must be odd polynomials with positive coefficients.
Let $m,n\geq0$ are integers, show that
$$p_{m,n}(x)=\sum_{k=0}^{m} \binom{2x+2k}{2k+1} \binom{n+m-k-x-\frac{1}{2}}{m-k}$$
must be an odd polynomial(all coefficients of even power of $x$ is $0$) with ...
- 134
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56
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$f_n(ab) = f_n(a)f_n(b) + f_n(a-1)f_n(b-1) + f_n(a-2)f_n(b-2)+ ...+f_n(a-n)f_n(b-n)$
I was toying around when I noticed for $a,b > 0$:
$$f(ab) = f(a)f(b) + f(a-1)f(b-1)$$
is satisfied by $f(n) = T_n$ ; the triangular numbers $n(n+1)/2$.
This equation is not an addition formula, not ...
- 16.4k
2
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70
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Expectation of a certain polynomial expression in Rademacher random variables.
Let $N_1,k \ge 1$ be integers and let $N = N_1 k$. Let $G_1,...,G_k$ be an equi-partition of $[N] := \{1,2,\ldots,N\}$. Thus, $|G_j| = N_1$ for all $i$. Let $\mathcal S$ be the transversal of this ...
- 9,575
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194
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Conjecture: Shallow diagonals in Pascal's triangle form polynomials whose roots are all real, distinct, and in $(-2,2)$.
Here are the first few "shallow diagonals" in Pascal's triangle.
We can use these shallow diagonals to make polynomials. Note that the even degree polynomials have an extra $\pm x$ at the ...
- 25.7k
5
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215
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Series of polynomials very nearly follows binomial coefficients but doesn't quite
I'm modelling a system using a Markov chain and by a few iterations of the transition matrix I can see a pattern emerging in the resulting polynomial that really looks like Pascal's triangle, but isn'...
- 53
1
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2
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153
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Find the coefficients of product
Given the following product,
$$(1+ax)(1+a^2x)(1+a^3x)\cdots (1+a^mx) $$
where $a$ is some real number which will be taken to be unity in the end. I want to know the coefficient of general term of ...
- 505
5
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169
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Prove that $\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$
For arbitrary $x$ and $1\leqslant m\leqslant n$, prove the following:
$$\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$$
I'm looking for a proof that ...
- 848
1
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1
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97
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Sum with binomial coefficient using identity
I want to prove:
$\displaystyle \sum_{k=0}^n (-1)^k \binom{x}{k} = (-1)^n \binom{x-1}{n}$
using: $(1-z)^x \cdot \frac{1}{1-z} = (1-z)^{x-1}$
I know how to do it with induction but i somehow can't ...
- 121
0
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0
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42
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Expressing $j$-nomial coefficients in terms of binomial coefficients
Call the $k$th coefficient of $\left(\sum_{i=0}^jx^i\right)^n$ the $j$-nomial coefficient $C(n,k,j)$, so that the numbers $C(n,k,2)$ are just the binomial coefficients $\binom{n}{k}$. From the OEIS ...
- 2,346
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51
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Show with the help of binomial theorem that these two equations are equal?
Show with the help of binomial theorem that these two expression are equal for $n\ge 0$ then this $$ \sum_{k=0}^n \binom n k x^k (2+x)^k = \sum_{k=0}^{2n} \binom {2n} k x^k $$
I don’t know how to do ...
- 51
7
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3
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275
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Proof of a neat pattern in polynomials
Let $f_1:\mathbb{R}\to\mathbb{R}$ such that $$f(x) = ax + b\space\space \forall\space x \in \mathbb{R}$$
It can be easily verified that $$f(x)-2f(x-1)+f(x-2)=0 \space \forall \space x \in
\mathbb{R}--...
- 271
3
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1
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889
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What is the most efficient algorithm to evaluate a polynomial of n degree at K points?
A brute force approach would be to evaluate each point for each of the terms of a polynomial, which will be $O(Kn^2)$.
If we use logarithmic exponentiation to find each $x^i$ then it becomes $O(Kn \...
- 491
1
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0
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50
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Nonnegative integer combinations of binomial coefficients
A classical result of Polya and Ostrowski states that integral linear combinations of binomial coefficients ${x\choose k}$ is exactly the set of all polynomials $f(x)\in\mathbb{Q}[x]$ such that $f(\...
- 427
2
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129
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Concrete Mathematics: What is polynomial argument
In the book Concrete Mathematics: A foundation for Computer Science of the chapter Binomial Coefficients, It gives an identity
$$
(r - k) {r \choose k} = r {r - 1 \choose k}
$$
for positive integers r....
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