All Questions
9
questions
0
votes
1
answer
51
views
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
2
votes
3
answers
184
views
Combinatorial interpretation of polynomials
I'm reading a proof for the following identity:
$$\sum_{k}A_{k}(r,t)A_{n-k}(s,t) = A_{n}(r+s, t), \qquad \text{integer} \ n \geq 0 \tag{1}$$
where $A_{n}(x,t)$ is the $n$th degree polynomial in $x$ ...
- 673
0
votes
1
answer
30
views
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
votes
4
answers
210
views
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
0
votes
1
answer
77
views
Finding a generating function
Suppose I have 4 numbers, $x_0,x_1,x_2$ and $x_3$, and the sum,
$$x_0+x_1+x_2+x_3$$
I put the constraint that $x_0$ and $x_3$ are either 1 or 0, and $x_0$ and $x_3$ can be equal or between $0$ to $3; ...
- 493
4
votes
1
answer
76
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Showing an identity between polynomials whose coefficients involve combinatorial identities
I want to show that
$$
\sum_{k=0}^{\lfloor n/2 \rfloor} \sum_{l=0}^{\lfloor \frac{n-2k}{2}\rfloor} (-1)^l \binom{n}{k} \binom{n-2k-l}{l} \frac{n-2k}{n-2k-l} x^{n-2k-2l} = x^n.
$$
If we "compare ...
- 18.2k
0
votes
2
answers
67
views
Way to find coefficient in the given expression
I need to find the number of distinct ways in which K unlabeled objects can be distributed in N labeled urns allowing at most 10 objects to fall in each urn i.e
Coefficient of $x^k$ in $(1 + x + x^{2} ...
- 335
3
votes
2
answers
1k
views
Proof of Pascal' identity
The identity
$$\binom{x+1}{k}-\binom{x}{k}=\binom{x}{k-1}$$
is claimed to hold (using the binomial polynomials, considered as lying in $\mathbf{Q}[x]$) for $k$ at least $1$.
Proof: by the usual ...
- 65
2
votes
2
answers
236
views
Expansion Coefficient needed
This is probably something very easy, but wth... my mind is totally stuck right now.
I need to find the coefficient of $x^{11}$ of the expansion $(x^2 + 2\frac yx)^{10}$
Well I know that the answer ...
- 23