Show that $\sum_{k=0}^{2n}(-1)^k \binom{2n}{k}^2 = (-1)^n\binom{2n}{n}$ Show that $$\displaystyle\sum_{k=0}^{2n}(-1)^k \dbinom{2n}{k}^2 = (-1)^n\dbinom{2n}{n}$$ where  $n \in \mathbb{N}$   Induction didn't work, and the formula $\displaystyle\sum_{k=0}^{2n}\dbinom{2n}{k}^2 = \dbinom{4n}{2n}$ wasn't very usefull. $(1+x)^{2n} = \displaystyle\sum_{r=0}^{2n} \dbinom{2n}{r} x^{r}$ and $(1-x)^{2n} = \displaystyle\sum_{l=0}^{2n} \dbinom{2n}{l} (-1)^lx^{l}$. Multiplying the two gives me,   $$ (1+x)^{2n}(1-x)^{2n} = (1-x^2)^{2n} = \displaystyle\sum_{k=0}^{2n} \dbinom{2n}{k}(-1)^kx^{2k} = \left( \displaystyle\sum_{r=0}^{2n} \dbinom{2n}{r} x^{r}\right)\left( \displaystyle\sum_{l=0}^{2n} \dbinom{2n}{l} (-1)^lx^{l}\right) $$  Coefficient of $x^{2n}$ on left side will be $\dbinom{2n}{n}(-1)^n$. But how do I exactly calculate the coefficient of $x^{2n}$ in the R.H.S? Please give a detailed solution
Proof/Derivation of Closed form of Binomial Expression $\sum\limits_{k=0}^{2n}(-1)^k\binom{2n}{k}^2$ The binomial expression given as follows: $$\sum_{k=0}^{2n}\left(-1\right)^{k}\binom{2n}{k}^{2}$$ results nicely into the following closed form: $$(-1)^{n}\binom{2n}{n}$$ I wish to know how exactly is it being done? I haven't been able to make much progress in solving it. My approach: \begin{align} \sum_{k=0}^{2n}(-1)^{k}\binom{2n}{k}^{2} = \binom{2n}{0}^{2} - \binom{2n}{1}^{2} + \binom{2n}{2}^{2} - ... -\binom{2n}{2n-1}^{2} + \binom{2n}{2n}^{2} \\ = \binom{2n}{0}.\binom{2n}{0} - \binom{2n}{1}.\binom{2n}{1} + \binom{2n}{2}.\binom{2n}{2} - ... -\binom{2n}{2n-1}.\binom{2n}{2n-1} + \binom{2n}{2n}.\binom{2n}{2n} \\ \text{By Symmetry of binomial coefficients} \\ = \binom{2n}{2n}.\binom{2n}{0} - \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} - ... -\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} \\ = \binom{2n}{2n}.\binom{2n}{0} + \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} + ... +\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} - 2.\left(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}\right) \\ \text{By Vandermond's identity, first component, i.e.not enclosed within -2.(...) evaluates to C(4n, 2n)} \\ \binom{4n}{2n} - 2.(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}) \end{align} I'm lost beyond this point. It will be extremely helpful if someone can direct me in the right direction or provide the answer to this perplexing and challenging problem. Thank you.
What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$  I tried to factorize $n^3+100$, but $100$ is not a perfect cube. I wish it were $1000$. Date Format In RUBY How to show $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$? Let $a\geq 0$ and $ b\geq 0$. Prove that $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$. [Hint: Use the identity $(a^n -b^n)=(a-b)(\sum_{i=0}^{n-1}a^ib^{n-1-i})$]    I need some help! I cannot do it even with the hint... :( I have come across both the phrases - 'on the internet' and 'over the internet.'   I'm pretty sure that both are correct but then which to use when?  Please note, I'm well aware of all over the Internet which I'm not concerned about. Suppose I look at a parallel plate capacitor in its rest frame and calculate the electrostatic energy, $E$.  Next, I look at the same capacitor in a primed frame boosted in the direction perpendicular to the plane of the plates. In this frame, the $E$-field is the same strength, there is no magnetic field, and the volume over which the $E$-field extends is less by a factor $1/\gamma$. This suggests $E' = \frac{1}{\gamma} E$, but relativity states that energy transforms as $E' = \gamma E$.   Where is the missing energy? Minipage: 2 tables side by side isn't working I have the following code to layout two tables side by side:  \begin{minipage}{0.3\textwidth} \begin{tabular}{c | c | c | c | c | c | c} F &amp; D &amp; E &amp; G &amp; P &amp; A &amp; L\\ \hline C &amp; H &amp; I &amp; I &amp; U &amp; G &amp; F\\ \hline G &amp; Z &amp; G &amp; A &amp; F &amp; Z &amp; U\\ \hline F &amp; R &amp; \textbf{H} &amp; \textbf{A} &amp; \textbf{U} &amp; \textbf{S} &amp; Z\\ \hline R &amp; P &amp; L &amp; I &amp; N &amp; F &amp; H\\ \hline L &amp; U &amp; G &amp; I &amp; A &amp; C &amp; T\\ \hline A &amp; B &amp; D &amp; N &amp; E &amp; R &amp; Z\\ \end{tabular}  \end{minipage}  \begin{minipage}{0.3\textwidth} \begin{tabular}{c | c | c | c | c | c | c} F &amp; D &amp; E &amp; G &amp; P &amp; A &amp; L\\ \hline C &amp; H &amp; I &amp; I &amp; U &amp; G &amp; F\\ \hline G &amp; Z &amp; G &amp; A &amp; F &amp; Z &amp; U\\ \hline F &amp; R &amp; \textbf{H} &amp; \textbf{A} &amp; \textbf{U} &amp; \textbf{S} &amp; Z\\ \hline R &amp; P &amp; L &amp; I &amp; N &amp; F &amp; H\\ \hline L &amp; U &amp; G &amp; I &amp; A &amp; C &amp; T\\ \hline A &amp; B &amp; D &amp; N &amp; E &amp; R &amp; Z\\ \end{tabular}  \end{minipage}   But it doesn't work. The tables are one below the other. What am I doing wrong? How do I list all files under a directory recursively and sort the output by modification time?  I normally use ls -lhtc but it doesn't find all files recursively. I am using Linux and Mac.  ls -l on Mac OS X can give  -rw-r--r--    1 fsr  user      1928 Mar  1  2011 foo.c -rwx------    1 fsr  user      3509 Feb 25 14:34 bar.c   where the date part isn't consistent or aligned, so a solution have to take this into account.  Partial solution  stat -f "%m%t%Sm %N" ./* | sort -rn | head -3 | cut -f2-   works, but not recursively. Create UEFI bootable USB stick to install Windows AND Ubuntu from it I know how to make a UEFI bootable stick to install Ubuntu (explained ) and I know how to make one to install Windows 7 (, sorry). I guess for Win 8 the procedure would be similar, but I haven't looked into it.  I also know how to make a legacy bootable stick from which I can install Windows and Ubuntu using YUMI. I have failed so far though at creating a UEFI bootable stick that can do this. When I do a UEFI boot from my YUMI multiboot stick it gives me no option to choose between Ubuntu and Windows (hardly surprising I guess) and instead simply UEFI boots Windows (which I find kind of surprising to be honest but maybe this is due to the fact that I added the Windows image after the Ubuntu image and this simply replaced Ubuntu's EFI boot files).  So how would I proceed to create such a bootable stick? Emacs: How to use a major mode for non-standard file extension Proof that in a field $x=0$ is equivalent to $x=-x$ Subscribers currently in a Journey What ways could you create a data extension that contains all the subscribers (SubscriberKeys) that are currently “in” a Journey (Journey Builder)?  The Journey I would like to do this for is a complex activity with multiple decision branches resulting in multiple possible journey durations. For example; if the customer didn’t open Email1 after 2 days, then send Email2; else Exit Journey. If the customer didn’t open Email2 after 2 days, then send Email3; else Exit Journey. Wait 2 more days, then End Journey. The minimum journey duration for the above example is 2 days, while the maximum is 6 days.  Note that I don’t want a list of subscribers that “entered” a Journey, but a list of the ones that are “currently in” a Journey and have not yet exited. Answer must account for non-standard Journey exits including Unsubscribes/Contact Deletes, etc. Merge pdfs sequentially I would like to merge 4 pdfs, pdf1.pdf, pdf2.pdf, pdf3.pdf pdf4.pdf together so that the pdfs are spliced, i.e. the first page will be the first page of pdf1, the second the first of pdf2, the third the first of pdf3, etc.  The pdf files are with plain images, no table of contents.  How would I accomplish this? It would be nice to do it from Terminal as well. Why don't applications update by calculating difference? What are the common phrases, words, abbreviations that are used on Stack Overflow, Server Fault, Super User, Meta Stack Overflow, and the other Stack Exchange sites? This is meant to be a very quick overview, not an in-depth tutorial. When considering whether a term ought to be included, please use this test:  New user comes in and posts on day 1. What terminology or jargon are they likely to run into (comments, support, system messages, etc.) that cannot be understood from the context without prior experience with the site (that is, clear only if known). For example: &quot;This is a dupe. Flag a diamond mod to close or migrate to SU.&quot; Huh? Can these instruction be made clear from the glossary? Is it clear for those for whom English is a second language?  Please:  Edit the existing answer Insert relevant links to more detailed information on the term/phrase/abbreviation Add only factual information Keep it very, very brief, terse, and to the point Delete your own comments once they become irrelevant (integrated into the glossary) Use the Edit Summary box to note reasons for your edit, not comments  Please avoid:  Opinion Using abbreviations, terms, and phrases inside a description Unverified information Uncommon terms, abbreviations, phrases Long descriptions Unnecessary detail - link to the relevant FAQ instead Citations or examples of usage Inside jokes that are not common on all of the trilogy () Common Internet jargon (that is, , , , etc.) unless usage here differs significantly  Recurring discussion: While the experiment to keep meta-discussion in the comments is going well (that is, deleting them once consensus is reached) it is clear that some discussions will be recurring, and we'll have to keep a record around so people new to the glossary can understand the consensus already reached. Add a new answer if you believe a new issue is likely to come up repeatedly. For instance, discussion about what terms to include, whether the glossary should be broken up to make linking easier, or formatting issues might be best discussed in new answer/comment posts. Optional Android notifications for reputation changes I would like to get a notification if my reputation count has changed. I think there is already some internal API for that. (I can see that changes in the menu bar in the top).  I'm sure that some people would not like that so that should be optional. Why did Tony Stark leave his armor to explore the castle in Sokovia? In Avengers: Age Of Ultron, Tony Stark is the first member of the Avengers to get inside Baron von Strucker's castle, where he takes down a room full of soldiers before stepping out of his suit and travelling down a secret passage.  Why does he take off his suit in the middle of an enemy base when it might have protected him against Scarlet Witch's magic, or at the very least helped him better search the passageway with Jarvis' HUD? Where can I find the location of folders for installed programs? With the enthusiastic question of "Who wants ice-cream?", what is the more correct response?         (Not) I.   (Not) me.      Neither response is a sentence. The first response of "(not) I" sounds stuffy, like it should be followed with an indignant sniff. The second sounds like American idiom and acceptable for casual speech.    What do you say? Partial Sums of Geometric Series How can I access and process nested objects, arrays or JSON?