Tagged Questions
23
votes
5answers
325 views
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$
Is there a closed form for the following infinite product?
$$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
11
votes
0answers
63 views
Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$
This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
0
votes
1answer
53 views
How to express this recurrence relation as a closed form?
I need a little help with expressing this recurrence relation as a closed form. I've already expanded it out to see the pattern:
$$
f(n) = f\left(\frac{n}{3}\right) + f\left(\frac{2n}{3}\right) + n - ...
5
votes
3answers
157 views
Calculate $\sum\limits_{k=801}^{849}{ \binom {2400} {k}} $
Is any formula which can help me to calculate directly the following sum :
$$\sum_{k=801}^{849} \binom {2400} {k} \text{ ? } $$
Or can you help me for an approximation?
Thanks :)
2
votes
2answers
66 views
Finding the expression for $q_n$
Let $q_n$ be the number of $n$-letter words consisting of letters a, b, c and d, and which contain an odd number of letters $b$. Prove that
$$q_{n+1} = 2q_n + 4^n\qquad\forall n \geq 1 $$
and, ...
2
votes
1answer
45 views
Recurrence equation question
My question (which has been edited) relates to the following recurrence relation:
$$a_{j+2}=\frac{2 a_{j}}{j}$$
The book which I am reading says that the (approximate) solution is given by:
...
2
votes
3answers
80 views
Please help solve the following recurrences
Please help with solving the recurrences to get closed form formulas for $a_n$, $b_n$ and $c_n$. Be sure to clearly label the characteristic equation, the roots of the characteristic equation, the ...
1
vote
1answer
185 views
No closed form for the partial sum of ${n\choose k}$ for $k \le K$?
In Concrete Mathematics, the authors state that there is no closed form for
$$\sum_{k\le K}{n\choose k}.$$
This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the ...
1
vote
1answer
136 views
Does a closed form formula for the series ${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$ exist.
$${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$$
For the above series, does a closed form exist?
1
vote
1answer
116 views
Looking for a closed form to determine whether a symbol is part of the ith combination nCr
Hi I'm new to this, feel free to correct or edit anything if I haven't done something properly.
This is a programming problem I'm having and finding a closed form instead of looping would help a lot.
...
