I was trying to prove some equation and reached this summation $\sum \limits_{j=0}^{m} {2j\choose{j}} {2m-2j \choose {m-j}} = 4^m $  I tried pascal identity and other known binomial identities but with no succes   Is there any simple proof (preferably algebraic one).  Thanks
It's not difficult to show that  $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$  On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$.  Squaring the first power series and comparing terms gives us  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}2^{-2n}=1$$  that is,  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$$  My question: is there a more direct, combinatorial proof of this identity?  I've been racking my brains trying to come up with one but I'm not having much success.
How to measure capacitance without LC meter? I would like to use capacitors salvaged from old electronics, but I currently don't have an LC meter.    What would be some really simple circuits, that would convert capacitance to something I can measure (voltage, current, frequency)? Error/warning when using NSolve for simple equation Lock keys panel indicator for MATE Desktop I have no indicator LEDs on my keyboard to show whether Caps-Lock is on (I don't have Num-Lock and Scroll-Lock keys so the best option would be to have an indicator only for Caps Lock, but three indicators would be better than none)  I would really like to see the lock-keys status in the indicator panel so I can tell at a glance whether I'm typing my passwords correctly    I'm using Ubuntu MATE 16.04. I've seen  but it relates to Unity DE. Protect Google Spreadsheet formatting while allowing changes to data I would like to protect the formatting of a cell (or a range of cells) in a Google Spreadsheet, but still allow changing the data in that (or those) cell(s).  For example, I want to add a validation rule on a cell that says the cell has to contain a number between 1 and 10, and I would like others to be able to change the value in that cell, but not the validation rule itself.  How do I do that?  So far I was only able to completely protect the cell (content + formatting) or not at all (the user can change the content as well as the formatting).  To me this seems to be an important functionality when adding access rights to spreadsheets. Is it possible that it is completely missing from Google Spreadsheets? Ellipsis for overflow text in dropdown boxes Is it okay to use my US visa after a name change? I had my US visa about 7 month age, and I had to renew my passport for certain circumstances, so there was a mismatch in the name between both the visa and the new passport, you see they had to put my grand grandfather's name which was not on the old passport nor the visa, the name on old passport and visa is: Abdelrahman Samy Ali Mohammed and the new passport is: Abdelrahman Samy Ali Ibrahim Mohammed. My grand grandfather's name is Ibrahim, so the question is that gonna cause me any troubles concerning my entry to the US? is there any document should I consider taking it while i'm traveling?  Thanks,, For any real number $a \geq 1$ let $f(a)$ denote the real solution of the equation $x(1+\ln x)=a$ then the    question is to find out $$ \lim_{a \to \infty} \frac{f(a)\ln a}{a}$$.  It is clear that if we denote $h(a)$ by $h(a)=a(1+\ln a)$ then $f(a)$ is the inverse function of $h(a)$. Also $f(a)$ is increasing function in its domain. Also the limit persuades using lhospital's but I cannot see how to apply it here. Thanks. Do you ever need to specify 'javascript:' in an onclick? AFAIK, you never need to specify the protocol in an onclick:  onclick="javascript:myFunction()" Bad  onclick="myFunction()" Good  Today I noticed in  on Google Anallytics that they are using it:  &lt;a href="http://www.example.com" onClick="javascript: pageTracker._trackPageview('/outgoing/example.com');"&gt;   Is this example just plain wrong, or is there ever a reason to specify javascript: in anything other than a href? Require reason for post deletion and un-deletion? Could we have a method of explaining why we think a question or answer should be deleted?  It could work identically to closing a question, maybe with an additional text-input box (like the "Requires Moderator attention" flag option) to provide additional info because the reasons for deletion should really be more specific..  For example, "exact duplicate" doesn't explain to me the user has re-posted the same question, thus, it should definitely be deleted (not just left closed). Like with close-messages, these reasons would be listed under the deleted item (or answer).   I don't think comments are suitable, as they can be a pain to quickly.. "extract" deletion reasons from them - I have to scan through the comments, and they may all be irrelevant.  Equally, for un-deletion, it'd be useful to know why another user thinks a question should be restored. For example "new user deleted the question after they accepted an answer", or "Jeff went insane and started deleting any questions with the letter O in it".  I think requiring un-deletion reasons is possibly more important, since commenting is disabled on deleted questions - we have no way of explaining why we think it should be restored For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of  $p$   So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to show that.Can anybody help me this method?Any other type of prove is also welcomed. In the movies, the pack of Uruk-Hai capture Merry and Pippin, and are bringing them back to Saruman as quickly as possible.  Why didn't they use Wargs to carry the Hobbits instead of packing them on their backs?    Or, better yet, why didn't the Nazgul fly down and take them on their Fell Beasts? Why does inertia happen? In  R P Feynman relates a story where his father told him that, even though we know the word "inertia" and what it means, nobody knows why inertia happens. Is that still true? Why I can't apply mirror modifier? Here's screen How do you remove Subversion control for a folder? I have a folder, c:\websites\test, and it contains folders and files that were checked out from a repository that no longer exists. How do I get Subversion to stop tracking that folder and any of the subfolders and files?   I know I could simply delete the .svn folder, but there are a lot of sub-folders in many layers. Reading a 5 Band resistor with Gold as a middle band? Can anyone read the resistor that the arrow is pointing to?  From what I know you can not have a gold band as a number and if I am reading it correctly it has a tolerance of 0 which does not seem right. Constants of motion vs. integrals of motion vs. first integrals Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of them, $2N-1$ are initial positions and velocity. Any function of these initial condition is a constant of motion, by definition. Also, there should be exactly $2N-1$ algebraically independent constants of motion.  On the other hand, Noether's procedure gives us integrals of motion as a result of variational symmetries of the action. These integrals of motion are also conserved but they are not always $2N-1$ in number. In consequence, we classify the system by their integrability.  So, what is the difference between the  and ? Why do non- have less integrals of motion when they should always have $2N-1$ constants of motion? Ubuntu 12.04 APT Proxy? I'm using Ubuntu 12.04 on a password-protected proxy network. My web browser is configured and works, but I cannot install packages from apt or similar. Is there a way to configure apt to use my proxy? Is there a way to configure my entire system to use my proxy? How can I keep track of a decimal number that indicates the current paragraph's place in a logical hierarchy and print it in the margin of each paragraph?  (To yield a numbering in the style of Tractatus Logico-Philosophicus).  \def\on{} \def\down{} \def\up{}  \on This is the first paragraph, paragraph number 1.  \on This is paragraph number 2.  \down This is paragraph number 2.1  \down 2 This is 2.101  \up This is 2.11  \down 2.111  \up 3 3  \bye   If \down is starred, begin with zero.  If \up is followed by an integer, it should take it as an argument, unless there is a newline between \up and the integer.  An indefinite number of decimals should be possible.  PS.  I apologize for changing the question after (2) answers have been given, but I realize I failed to appreciate a feature of Wittgenstein's numbering: the last decimal is never 0.  Therefore, \down &lt;n&gt; is better than \down*, as now demonstrated above.  I think the 0 is supposed to indicate an additional level of unimportance. Consider a partial function $f$ that is defined only for a few values of its domain (my exact use case is $\delta$ transition functions in automata). One can 'complete' it by saying $$g(x)=0\iff f(x) \text{ is not defined.}$$  Is there a symbol to mean "undefined"? Would it be correct, or accurate, to write $\nexists f(x)$? I'm running Ubuntu server on a computer used as a wireless AP, but this AP should resolve all DNS requests to an internal IP address rather than actually performing the lookup.  I want to do the same thing that paid public WiFi hotspots do - you can connect but if you attempt to load any websites they show a default page. I've noticed that they do this by resolving all domains to an internal IP address.  I've added these lines to /etc/dnsmasq.conf:  # Add domains which you want to force to an IP address here. # The example below send any host in double-click.net to a local # web-server. address=/com/192.168.2.1 address=/uk/192.168.2.1 address=/org/192.168.2.1 address=/gov/192.168.2.1 address=/net/192.168.2.1 address=/us/192.168.2.1   which works fine for those TLD's, but I'd like to be able to do it with all domains so I can sleep at night.