Tagged Questions
0
votes
1answer
39 views
What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$
For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series
$$
\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n.
$$
For $b=0$, this post shows
$$
\sum_{n=0}^\infty ...
4
votes
2answers
73 views
Finding a closed form expression for this sum [duplicate]
For non-negative $n$, find
$$
\sum_{k=0}^n \binom{2k}{k}\binom{2n-2k}{n-k}.
$$
I can't figure this out. Any ideas?
0
votes
1answer
88 views
Using the general binomial theorem to find a series-like expression for $\sqrt 2$
How do I use the general binomial theorem (i.e. the series expansion of ${(1+x)^\alpha}$ for $ |x|<1$) to show the following? $$\sqrt 2=1+\frac 1{2^2}+\frac{1\cdot3}{2!\cdot{2^4}} ...
1
vote
2answers
48 views
Integer sequence comparison, binomials Vs power
I need to know which sequence grows faster with n:
$$ f(n) = \sum_0^{floor(n/3)} {n \choose 3*i+1} $$
compared to
$$ g(n) = 2^n -1 $$
it seems f(10)>5000 is greater than g(10)=1023 but I would ...
3
votes
1answer
181 views
A sum involving permutation
Does there exist a nice closed form formula for the sum
$$\sum_{k=0}^m P(m,k)x^k$$
where $P(m,k)=C(m,k)*k!$, $C(m,k)$ being the "m choose k" number.
Formula given by Maple 11 is complicated. I ...
4
votes
2answers
492 views
Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$
How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$?
I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me.
(This series arises in ...
