0
votes
1answer
58 views

Closed form for nth term of generating function

How would I find the closed form for the $n^{th}$ term of a sequence? Is there a general formula I can follow for these types of problems? Taking this sequence for example... $$\frac{x^5}{(1-x)^4}$$
5
votes
2answers
72 views

Closed form of generating function consisting of power of two binomials

Let $g(x)$ be infinite formal power series and $$g(x) = (1 + x)(1 + x^2)\cdots(1 + x^{2^k})\cdots$$ Show that $(1 - x) g(x) = 1$. My book gives following proof: Using a fact that $(1 - x^k)(1 + ...
1
vote
3answers
214 views

Is there a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$?

I am interested in finding a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$. Does anyone know if there is some Binomial identity that might be helpful here? Thank you.
1
vote
1answer
105 views

Recursive and closed form solution for choosing $n$ pairs/triplets.. of $kn$ elements.

I stumbled apon an interesting question: How many ways are there to arrenge $kn$ elements into $n$ sets, $k$ elements each? There should be a recursive and closed form solution for $g_k(n)$. For ...
4
votes
1answer
113 views

Is there a closed form for $\sum_{i=0}^{\text{min}(k,t)} {{\binom k i} \over {\binom t i}}\cdot x^i$?

Is there a closed form for the following summation? ($k,t \in N, x\in \mathbb R^+$) $$f(x,k,t)=\sum_{i=0}^{\text{min}(k,t)} {{\binom k i} \over {\binom t i}}\cdot x^i$$
12
votes
2answers
309 views

Determining the number of valid TicTacToe board states in terms of board dimension

I am attempting to find a closed form equation in terms of $n$, for the number of valid Tic-Tac-Toe board states (ignoring symmetry), where the board has dimension $n \times n ,\; 0 \lt n,\;n \in \Bbb ...
33
votes
5answers
667 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? $$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
22
votes
1answer
494 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
101 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
200 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
77 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
52 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
103 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 ...
2
votes
1answer
298 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
172 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
2answers
181 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. ...