All Questions
Tagged with binomial-coefficients polynomials
188
questions
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Show that $P_n(X)=\frac{(X+i)^{2n+1} - (X - i)^{2n+1}}{2i}$ is of degree $2n$, even..., using the binomial coefficients formula [duplicate]
In the previous questions i've proved that $(1+i)^{2n+1}=a_n+ib_n$ where $a_n$ and $b_n$ are $\pm 2^n$ and that $(1-i)^{2n+1}=a_n - ib_n$ and that $|P_n(1)|=2^n$ using the previous statements. But now ...
- 143
1
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1
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174
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What is the constant term in the expansion of ${\left(x^2-\dfrac{1}{x^4}\right)}^{13}$ ? and the middle term in it?
I want to determine the constant term and middle term in the expansion of ${\left(x^2-\dfrac{1}{x^4}\right)}^{13}$, I know that the constant term in the expansion of $(x+a)^n$ is always $a^n $, We can ...
6
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4
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258
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If $P(x)$ is any polynomial of degree less than $n$, show that $\sum_{j=0}^n (-1)^j\binom{n}{j}P(j)=0$. [duplicate]
If $P(x)$ is any polynomial of degree less than $n$, then prove that
$$\sum_{j=0}^n (-1)^j\binom{n}{j}P(j)=0$$
My approach was to try and prove this separately for $j^k\ \ \forall\ \ k<n$, instead ...
- 641
3
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1
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56
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Find all the values of a so that $3^{ \lfloor \frac{n-1}{2} \rfloor }\mid P_n{(a^3)}$ given the definition of $P_n$
Consider the polynomial $P_n{(x)} = \binom{n}{2}+\binom{n}{5}x + \cdots \binom{n}{3k+2}x^{k}$, where $n \ge 2$, $k= \lfloor \frac{n-2}{3} \rfloor$,
$(1)$ Show that $P_{n+3}{(x)}=3P_{n+2}{(x)}-3P_{n+1}{...
- 1,288
1
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0
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71
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Why do two ways of expanding the same formal polynomial lead to matching coefficients?
There are a few proofs in which the technique is to expand the product of some formal polynomials in $\mathbb{R}[x_1,x_2,\ldots,x_k]$ in more than one distinct way and then we can match up the ...
- 3,415
2
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55
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Formula for $\sum\limits_{k=1}^m\binom{N}{k}k^n$ [closed]
For given natural numbers $m, n$ and $N$, is there a compact formula for the expression
$$\sum\limits_{k=1}^m\binom{N}{k}k^n\; ?$$
We can assume that $N>m$.
- 43
2
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241
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Monomials in terms of binomial coefficients
Is there an explicit expression (or at least a recurrence relation) for the coefficients of a monomial $x^n$ in the basis of polynomials given by binomial coefficients $P_k(x) = \binom{x}{k}$, namely
$...
- 63
2
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1
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101
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Proove that Abel polynomials are of binomial type
There are many places where the statement that these polynomials are of binomial type but no one actually proves that
Let
$$
P_n(x) = x(x + an)^{n-1}
$$
Prove that
$$
P_n(x+y) = \sum_{k=0}^{n}C_n^{k}...
- 23
2
votes
2
answers
80
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On Hilbert polynomials
The sequence of Hilbert polynomials is defined by :
$$H_0=1,\ \\ H_n(X)=\frac{X(X-1)...(X-n+1)}{n!}\ ,n>0$$
How can we prove the following identity :
$$H_n(X+Y)=\sum_{k=0}^nH_k(X)H_{n-k}(Y)$$
- 357
1
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1
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37
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A pattern with third coefficients of sums of powers.
So I have heard countless times of Bernoulli Numbers and its relation to the sums of powers. Inspired by this (as well as Chebyshev polynomials), I decided to look at the sums of powers myself and ...
- 9,487
5
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1
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398
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Reference request - identity in central factorial numbers
Knuth (in arXiv:math/9207222 [math.CA], page 10) gives an odd polynomial identity like
$$n^{2m-1} = \sum_{k=1}^{m} (2k-1)! T(2m,2k) \binom{n+k-1}{2k-1},$$
where $T(m,k)$ is cental factorial numbers....
- 558
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1
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30
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Is there a generalized formula for recusive Binomial coefficient?
Is there any way I can solve the following recursion function?
$$f(n) = \binom{f(n-1)}{2}$$
Or can be written as $$f(n) = 1/2(f(n-1)(f(n-1) - 1))$$
f(0) = 4, f(1) = 6, f(2) = 15, ...
- 367
2
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1
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228
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Expansion of $(a+b+c+d+e+....)^n$, but with all coefficients equal to 1.
I'm looking for a formula to calculate the sum of $(a+b+c+d+...)^n$ but with coefficients equal to 1.
For example in $(a+b+c)^2$. I want the sum of $a^2 + b^2 + c^2 + ab + bc + ca$. And for $(a+b+c+d)^...
- 123
0
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2
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51
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The Polynomials $P_n(x+y)$
Let $$\displaystyle P_n(x) = \sum_{k=0}^n \binom{n}{k}x^k.$$
We need to show that $$P_n(x+y) = \sum_{k=0}^n\binom{n}{k}P_k(x)y^{n-k}.$$
In the proof, we have $$\begin{array}{rcl}
P_n(x+y) &=&...
- 11
2
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4
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210
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Finding a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$
So the task is to find a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$
I was wondering if there is a more intelligible and less exhausting strategy in finding the coefficient, other ...
- 515