How can I prove the following identity: $$\sum_{0\le i\le n} (i+1)2^i {{2n-i}\choose{n}} =(2n+1){{2n}\choose{n}}$$ A few words as to why I need the proof. I noticed that if we write numbers of the form $n2^{n-1}$ (so, 0, 1, 4, 12, 32 etc.) on the sides of Pascal triangle and continue to build Pascal triangle by the same rule as usually, then numbers $n{{2n}\choose{n}}$ (2, 12, 60, etc.) emerge on the diagonal.  Here is the picture showing ratios of the numbers in the new triangle to the corresponding numbers in the usual Pascal triangle  One can write down the generating function with two variables for the entries in our new triangle. Let $a_{i,j}$ be the entry in the i-th column and j-th row, then $$F(x,y) = \sum a_{i, j} x^i y^j = \frac{ \frac{x-x^2}{(1-2x)^2} + \frac{y-y^2}{(1-2y)^2} }{1-x-y}$$ From here I obtained the aforementioned identity comparing the coefficients of $x^n y^n$. But I don't know how to prove it.  The upper coefficient in the sum depends on $i$, so this formula has no algebraic meaning. Induction doesn't help too. Any hints are welcome.
I used WolframAlpha to calculate a sum but it didn't show me the way :( Anybody has a hint or a solution for proving this sum? $$\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$$
Find all entire functions such that $|f(z)|\leq |\sin(z)|$ for all $z\in\mathbb C$. Why was "An Update to our Community and an Apology" deleted? Recently, Sara Chipps posted her message . After reading it, I wanted to compare it with . It turns out that she deleted it. Does anyone know why? It looks strange since (as far as remember) it was a sort of official position of SE. Why would such a statement be deleted? I have  today and one member was voted as duplicate and private feedback was there, I choose no and submitted the form. When I revisit the question or refresh the page same notice reappears with yes/no options.  In-network tab I got 200 OK status and response as "success":true  Environment: OS: Windows 10 Browser: Chrome 79.0.3945.117 (Official Build) (64-bit)  From the comments, the same issue is happing with   (might be with whole StackExchange network)      From network tab Proof of "the continuous image of a connected set is connected" None of the existing questions is exactly answering my question so I'm posting a new question, but feel free to refer me to some already answered question!   In Rudin Theorem 4.22, we know that      If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, and $E$ is a connected subset of $X$, then $f(E)$ is connected.    In the proof, we started with consider $f(E) = A \cup B$, where $A$ and $B$ are nonempty separated subsets. Then put $G = E \cap f^{-1}(A)$ and $H = E \cap f^{-1}(B)$. Then Rudin is claiming that $E = G \cup H$. I'm a little suspicious about this. What if $f$ is non-surjective, then $f^{-1}(A) \cup f^{-1}(B)$  is only a proper subset of $E$? Is there a property of $f$ being continuous that forces $f$ to be 1-1? Unknown apps/processes with number only are shown in data usage A mysterious app named &quot;10102&quot; shows up in my data usage. It bears only the standard Android app icon (the Bugdroid logo with a cube on its torso). From what the graph tells me, it has used 36 megabytes of data in the background and zero on the foreground. I think I haven't uninstalled anything for months. Is it malware? I haven't sideloaded anything and only authorized my laptop as a USB debugger. If it's malware, how do I remove it? I have no idea how to find its package name and 10102 isn't in the list of downloaded apps in settings. I'm running stock &amp; never-been-rooted Android 4.4.4 on Nexus 4. Which singular names ending in “s” form possessives with only a bare apostrophe? Many questions already ask about this topic ( , , etc.) and their answers vary, but they always give exceptions to the apostrophe-s rule, for example:      6.24 The general rule for the possessive of nouns covers most proper nouns, including most names ending in sibilants."   Examples they give include Kansas’s, Ross’s land, and Jones’s reputation. Exceptions include Jesus’  and Moses’.  Which names does this apply to?  Is the Aeneas’  form correct, or is it Aeneas’s instead? I am trying to run a function from  in ArcMap 10.1; Sink evaluation. But before it can finish it returns an error message, that, frankly, I don't understand at all.  It goes System.runtime.interopservices.comexception (0x80004005): error hresult e_fail .  Can anyone help me solve this? I have tried all I can think of. Run the function from the toolbox instead of dropdown menus, shorten paths, restart ArcMap, reinstall AHT, open new map documents, restart my computer, wait till next day and see if it automagically have fixed itself...   After searching this forum I found a , although it's got no answers. Now I'm all out of ideas, can anyone please help me?!    I'm using ArcMap 10.1 SP1 basic license and Windows 7.    I think I have narrowed it down a bit, still not clear what's wrong though. Sink evaluation uses a series of functions in Arcmap, of which one is Zonal statistics as table. That's the one it gets stuck on. The input raster somehow does not exist, path unknown. I guess this is some intermediary raster.       After consulting the Arc Hydro team, I got the advice not to use grids stored in a gdb (which I didn't). I have also tried to change target locations, thanks to @G.Oxsen. Unfortunately neither of those solved the issue. According to the Arc Hydro team, GPL0 that I asked about earlier is an in-memory grid, so probably no problem there.   I am currently pursuing that it is related to the installation of SP1. Anyone out there using AHT with SP1 and got Sink evaluation working?    Uninstalling SP1 did not change anything, unfortunately. Still working on this, but with less intensity. Seems like I'll have to find workarounds...    By reducing the number of sinks in need of processing I managed to get Sink evaluation running. Having the files not stored in a gdb seem to help as well, as mentioned earlier, and using integer grids.  Changing the "Maximum number of unique values to render" in ArcMap Options (Customize menu), tab Raster --> Raster dataset is another tip that I got (from the Arc Hydro Team). Restarting ArcMap after running every one or two functions also helps keeping things stable. Why isn't there a potential difference across a disconnected diode? I know this question sounds silly, as if there was a potential difference a current would be created when the terminals are connected together and this would mean energy has come from somewhere.  The reason I ask this though is that from my understanding of the depletion region and built in potential of a diode it seems like if you connected a voltmeter across the whole diode it would show the value of the built in potential.  This is explained in the image below:    At first, electrons flow from the n type to the p type because there are a higher concentration in the n type, and holes do vise versa. This is called the diffusion current. The first electrons and holes to cross the pn boundary are the ones which are closest to it; these carriers recombine when they meet each other and are then no longer a carrier. This means there is a depletion region of no carriers near the pn boundary. because electrons have left the n type material, and holes have left the p type material, there is a surplus of positive and negative charge on the n and p side of the pn boundary respectively. This causes an electric field that opposes the diffusion current, and so no more electrons or holes cross the boundary and combine. In short, only the electrons and holes near the boundary combine, because after they have done that an electric field is formed that prevents any more carriers from crossing. The current due to this electric field is called drift current, and when in equilibrium this will equal the diffusion current. Because there is an electric field at the boundary (pointing from the positive charge to the negative charge) there is an associated voltage. This is called the built in potential.  If you sample the electric field at each point along the diode from left to right, you would start with 0 in the p region because there are an equal number of protons and electrons. As you approach the depletion region you would see a small electric field pointing back towards the p region, caused by acceptor impurities which now have an extra electron (due to recombination) and therefore now have a net negative charge. This electric field would increase in strength as you get closer to the boundary, and then die away as you get further away.  This electric field means there is a voltage, as shown in graph (d). The p side is at an arbitrary potential, and the n side is at a potential higher than this because there is an electric field between them. This means there is a potential difference across the depletion region; this is known as the built-in potential.  But why when I connect a volt-meter across the whole diode will I not see this built in potential? I am used to see timestamp everywhere by it's color. The recent timestamps are a little bit reddish everywhere. E.g. in Meta Stack Overflow:  Last edit:        Last seen:        But in Meta Stack Exchange they have same color everywhere even for recent time:  Last edit:        Last seen:         Please make it something like this: Does anyone know how to remove this dead bolt that has no screw heads to use a screw driver? Here's a picture below. Word for dismissing someone's opinions as racist, liberal, etc, instead of debating back I'm looking for either a single word or phrase that would describe either someone or the action of dismissing someone's opinions as something "socially unpopular", without giving any reasoning why, either just to spite them, because they have a poor counter-argument, have been backed into a corner, or something similar.  For example, someone who supports something politically incorrect (like, say, stronger gun control or help for male domestic abuse) might have their views dismissed as racist or sexist or liberal without actually acknowledging and debating against them, thus labeling the former as something they might not be.  I was talking to a friend earlier about politics, and she complained about Germany's immigration problem. She said one issue was that many politicians that try to support stronger immigration laws would immediately be shot down as racist or a nazi and forced to resign, instead of being presented with counter-arguments. Regardless of whether you agree with them or not, what would describe that?  EDIT: Forgot to specify -- the word or phrase I'm looking for would be used in an informal context. And to address further questions, to be used when the person's opinion has legitimate worth and isn't just "being racist" etc, i.e. the dismisser is incorrect, and is only making these accusations because they can't argue further. (Added emphasis to the last sentence of the first paragraph) What does "...from nonmagical attacks not made with silvered weapons" mean? How to list all installed packages I'd like to output a list of all installed packages into a text file so that I can review it and bulk-install on another system.  How would I do this? alias fo rm command ( cp comand issue ) How to check who owns the Slack's workspace? I'd like to check who owns (or at least some kind of public profile page) some workspace based on its subdomain (e.g. xyz.slack.com).  Is it possible? The $n$ complex $n$th roots of a complex number $z$ Unitary map between sets of vectors If $\mathrm{Tr}(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$ In most electrical circuits with a relay, a diode is connected in parallel to the coil of the relay. Why? Is it always a good practice? Blender 2.78 with Windows 10 lags when interacting with the UI Update  This has been confirmed resolved in a later version 376.33 and possibly the update prior.     Okay, I recently did a computer update. I used to run Blender 2.77 on a Dell desktop (5 GB RAM, 2.7 GHZ, Windows 8, build in graphics card). Since that computer wasn't fast enough for my preference, I saved up and got a new gaming laptop - the Asus ROG GL552 (Windows 10, 16 GB DDR4 RAM, Nvidia GeForce GTX 960M with 4 GB DDR5 RAM, 2.6-3.5 Ghz, and quad-core intel processor). That laptop has been known to have outdated drivers, so when I got it, I updated it to up to date Windows 10 drivers.  As soon as I started Blender, I immediately noticed its speed. Even doing simple things, like clicking buttons, or rotating the view with the middle mouse button, took too long to register in the program. I tried changing some User Preference System settings, but that didn't really help much. And the hardware seems to be fine - the render is lightning fast (renders the startup scene in under .1 second; it's just laggy when loading the uv screen), so I don't know why it's like this, or what to do. Also, I ran Blender 2.77 on this computer, and it was also slow (although it was fine on my previous computer).  It could be an issue with drivers perhaps (or maybe OpenGL?), but I ran Unreal Engine 4, and had none of the same issues - the speed was pretty decent on that; so I'm not sure why Blender would be so slow. Any help is appreciated - thanks guys!