Using Vandermonde's identity, I can have $$ \sum_{i=0}^{n} \binom{n}{i}^2=\binom{2n}{n} $$ Is there a closed expression for $$ \sum_{i=0}^{n} \binom{n}{i}^2 x^{i}. $$ If we define a function $$ f(x)=\sum_{i=0}^{n} \binom{n}{i}^2 x^{i} $$ Then $f(1)=\binom{2n}{n}$. Can the expression of $f(x)$ be simplified?
122819
I've got the following finite sum: $s_{n}=\sum\limits_{k=0}^{n}\binom{n}{k}^2p^k$ (esp. if $p$ is a function of $n$, like $p=\frac1{n}$), which can be rewritten as $s_{n}=\sum\limits_{k=0}^{n}\binom{n}{k}\sqrt p^k\binom{n}{k}\sqrt p^k$. Using the generating function approach (from Graham, Knuth and Patashnik) with each of the polynomials in the sum, I get the expression $(1+\sqrt p x)^{2n}$ and the coefficient at the n-th term $x$ turns out to be $\binom{2n}{n} \sqrt p^n$. But comparing this result to the computational value, it turns out to be incorrect. Where did I make a mistake? Are there certain limitations on the generating function method?
101345
I tried getting it into a closed form but failed. Could someone help me out?  $$\sum_{n=0}^{\infty} \dfrac{a^n}{(3n)!}$$
100910
I was stuck while finding the given summation.  $$\sum_{k=1}^rk^2\binom {n-k}{r-k}$$  Since $n$ and $r$ are both constants, so I have first converted the above summation into this:  $$\sum_{k=1}^rk^2\binom {n-k}{n-r}$$  I have no idea how to proceed next. Any help will be appreciated.
78353
How can I calculate the following sum involving binomial terms:  $$\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$  Where the value of n can get very big (thus calculating the binomial coefficients is not feasible).  Is there a closed form for this summation?
89125
I was assigned the following problem: find the value of $$\sum_{k=1}^{n} k \binom {n} {k}$$ by using the derivative of $(1+x)^n$, but I'm basically clueless. Can anyone give me a hint?
104668
I need to prove this $$\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$$ Using induction and basically only the Pascal's identity and the symmetry property for binomial coefficients. Now, I noticed that $\sum_{k=0}^n \binom{r+k}{k}=\binom{r+n+1}{n}$ has a relatively simple proof by induction and it basically holds for any $r$. Could I apply that property for $r=n$ and call it a day? I also tried to solve the problem by using a similar argument, the induction step however states that $$\sum_{k=0}^{n+1} \binom{n+1+k}{k}=\sum_{k=0}^{n} \binom{n+1+k}{k}+\binom{2n+2}{n+1}$$ then $$\sum_{k=0}^{n+1} \binom{n+1+k}{k}=\sum_{k=0}^{n} \binom{n+k}{k}+\sum_{k=0}^{n} \binom{n+k}{k-1}+\binom{2n+2}{n+1}$$ Using the induction hypothesis $$\sum_{k=0}^{n+1} \binom{n+1+k}{k}= \binom{2n+1}{n}+\sum_{k=0}^{n} \binom{n+k}{k-1}+\binom{2n+2}{n+1}$$ And that's where I got stuck. Is this a dead end?
28782
I am trying to obtain a closed form expression for  $\sum {n \choose 4k}$  I am trying to use the binomial expansion of $(1 + i)^n$ and $(1 - i)^n$. $$(1 + i)^n + (1 - i)^n = 2\left(\sum {n \choose 4k} -\sum {n \choose 4k + 2}\right)$$ Stuck at this now. I can't come up with a way to simplify the second term on RHS. Any help would be appreciated.
8310
After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.
49298
I'm trying to prove that  \begin{equation} \sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1} \end{equation}  with the Binomial Theorem.  I know that the B.T. states that  \begin{equation} (x + y)^n =  \sum_{k= 0}^{n} \binom{n}{k} x^{n-k}y^{k}  \end{equation}  The proof that I have tried starts with:  \begin{equation} n \cdot 2^{n-1} =  \end{equation}  Actually I am stucked with this step.
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How do I prove the following identity:  $$\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$$  I am trying to use inclusion-exclusion, and this will boil down to a sum like inclusion-exclusion, and the $\binom{2n-2k}{n-1}$ term wouldn't matter (it will be equivalent to set sizes). Is this a correct way to go?
6864
I know that the sum of squares of binomial coefficients is just ${2n}\choose{n}$ but what is the closed expression for the sum ${n\choose 0}^2 - {n\choose 1}^2 + {n\choose 2}^2 + \cdots + (-1)^n {n\choose n}^2$?
43096
Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer.  I tried using binomial identities, but since the binomial coefficient is in the denominator, I can't do much. Also, I'm seeking a non-calculus, elementary solution, if possible. Thanks.
10141
Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$;  I was stuck with this question for a while... Help me please!!! Thanks!!!
37351
What is the closed formula for finding      $$ \sum_{i=0}^{k} \binom{n}{2i}=?\quad ,k\leq \lfloor{n/2}\rfloor $$   I need summation only up to some intermediate  $k$ where $k\leq \lfloor{n/2}\rfloor$ .  SAY I need a closed formula to calculate $\binom{8}{0}+ \binom{8}{2}+\binom{8}{4} +\binom{8}{6}$  HERE $n=8$ and $k=6$
16105
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 studying the first passage time of a simple random walk.)
104982
Prove the identity  $$\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$$  Use a combinatorial interpretation of the positive and negative terms.   Edit: I have already proven it using the binomial theorem, more specifically with $(1-1)^n$. I'm having difficulty understanding what is meant by a combinatorial interpretation of the positive and negative terms.    I'm having trouble understanding exactly what is necessary for a combinatorial interpretation. Looking at small cases, I see that the symmetry of the expansion allows us to cancel out lots of terms.   Is this what is meant by a combinatorial interpretation? Is there any resources that introduce the idea of what constitute a combinatorial interpretation?  I feel like the professor threw us off in the deep end and I'm trying to keep my head above water with the few problems I have to work with and that I understand.
109085
How can you show that $$ \binom{2n}{n}^2 = \sum_{m=0}^{n} \binom{2n}{2m} \binom{2m}m \binom{2n-2m}{n-m} $$?  I was fooling around with random walks, and apparently both expressions are supposed to be the number of unweighted random walks on a 2D square lattice of $2n$ steps that ends at the origin.  Whole thing reminds me of Vandermonde's identity but I can't it. Any hints?