Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly" This is a homework problem. In the first part of the problem, I managed to use a combinatorial problem to prove the following identity:  $\sum_{k=0}^{n}(-1)^k {2n-k \choose k} 2^{2n-2k} = 2n+1$  But I actually have problem with the second part of the problem which asked me to prove this identity "directly", probably using some form of generating functions/algebra method.  There was a hint given but I got stuck: The hint says to calculate   $\Sigma_{n=0}^\infty\Sigma_{k=0}^{n}(-1)^k {2n-k \choose k} 2^{2n-2k} x^{2n}$  and that it would be useful to consider the fact that $ \Sigma(2n+1)x^{2n}= \frac{d}{dx}\frac{x}{1-x^2} $ and $(1-x)^{-a-1}=\Sigma_{j=0}^\infty {a+j \choose j}x^j$.  I am able to prove both these "possibly useful facts" in the hint, but I don't see how to calculate the suggested double sum or prove the identity. [I would have thought that the combinatorial proof is harder to find!] [You may assume that I am familiar with the formal treatment of power series.]  Any help would be greatly appreciated.
Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$ I am dealing with problem 10F in the textbook A Course in Combinatorics by J. H. van Lint and R. M. Wilson.   Problem 10F: Prove directly the equality  $\sum\limits_{k = 0}^n {{{\left( { - 1} \right)}^k}\left( {\begin{array}{*{20}{c}} {2n - k}\\k\end{array}} \right){2^{2n - 2k}} = } 2n + 1$.  Sketch of the proof: Evaluate  $\sum\limits_{n = 0}^\infty  {\sum\limits_{k = 0}^n {{{\left( { - 1} \right)}^k}\left( {\begin{array}{*{20}{c}}{2n - k}\\k\end{array}} \right){2^{2n - 2k}}{x^{2n}}} } $. To do this, use Hint 1. Then, use Hint 2.  Hint 1: $\sum\limits_{j = 0}^\infty  {\left( {\begin{array}{*{20}{c}}{a + j}\\j \end{array}} \right){x^j} = {{\left( {1 - x} \right)}^{ - a - 1}}} ,\,\,a \in {\mathbb{Z}^ + }$.  Hint 2:  ${\sum\limits_{i = 0}^\infty  {\left( {2i + 1} \right){x^{2i}} = \left( {\frac{x}{{1 - {x^2}}}} \right)} ^\prime }$.   Attempt: It is enough to show that  $\sum\limits_{n = 0}^\infty  {\sum\limits_{k = 0}^n {{{\left( { - 1} \right)}^k}\left( {\begin{array}{*{20}{c}} {2n - k}\\ k \end{array}} \right){2^{2n - 2k}}{x^{2n}}} }  = {\left( {\frac{x}{{1 - {x^2}}}} \right)^\prime }$. But I can not use Hint 1.
Use chown to set the ownership of all a folder's subfolders and files? How to change default user in WSL Ubuntu bash on Windows 10 I installed the WSL Bash on Windows 10 and set my user to e.g. 'abc' but now I want to change it to something else.  How can I change the user name of the user I'm logged in as?  When I open bash it automatically logs me in with my current user. Can we build capacitors on a PCB board? One of my columns is called from. I can't change the name because I didn't make it. Am I allowed to do something like SELECT from FROM TableName or is there a special syntax to avoid the SQL Server being confused? why do academics write peer reviews? How are the best director award winners chosen? I do not understand how the best director awards are determined each year. It seems impossible to assess the method of a movie director unless you are part of the crew.  So, how do the nominators/judges assess the directors' performance each year? Seeking open source solution to access rasters in File Geodatabase? I'm looking for a way to access raster layers in an Esri File-GeoDatabase without using ArcObjects or ArcPy, specifically I would like to access the raster layers with GDAL. I understand that the GDAL FileGDB and OpenFileGDB give access to vector datasets within File-GeoDatabases, and that ESRI's own FileGDB C++ API does not support raster access.   Is anyone out there aware of any open source solutions? Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials.  We have that $R[[x]]$ is a ring containing $R[x]$ as a subring.  How to prove that if $a_0\in R$ is a unit, then  $\displaystyle\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$? This girl that I'm friends with and have actually seen in person before, keeps posting [on social media] that she "loves when the transfer goes through" and gets $2000-$6000 for free quite often. She said you can make up to $6500 a day. She is not very smart and barely spells correctly. She keeps saying that all she is doing is "transfers" and that's it. So many of my friends are feeding into this crap. So I asked her what it is. She basically said "it's legit" and asked me for my bank username and password so she could see how much money to deposit into my account. I said no way and left it at that. She also said she is not stealing or doing anything illegal. She is getting her friends free money and I think they're naive. What is she actually doing? Is it illegal? I am sure she is incapable of doing crazy money wiring to different countries.   Thank you everyone for answering my question. I am not stupid, I did not give her my information. I knew she was up to something illegal, so thank you for giving me some input on what exactly she is doing. I submitted an anonymous tip to my local sheriff's department. I'm hoping they see it and pursue it, or forward it to the right authorities I.e the FBI or something. setting up subdomain on ubuntu server Remove files with 0 length inside bash I want to create a bash script inside Linux terminal to go into a file and then delete it if it's empty:  #!/bin/bash echo " Enter name of a directory" read Dir for Directory in $( ls $HOME ) do    test -d $HOME/$dir   if [ $# -eq 0 ]   then      for filename in $( ls $HOME/$Dir )     do        test -d $HOME/$dir/$filename       if [ ! -s "$filename"  ]       then         rm $HOME/$dir/$filename       else          echo " not empty"       fi      done    else      echo " $Dir is not a directory"    fi  done   This is the code I am using but it does not work. It recognizes the file created but doesn't delete it even when empty. It Salah displays there is no such file or directory What is the optimal order of post-processing steps? How do you kill a java.lang.Thread in Java? I have a cheap Chinese made tablet from Dubai called the Crony Dubai (www.cronydubai.com). It has a micro B, mini HDMI and a USB labeled as "Host" as well as a micro SD slot. It came with a micro B to USB adapter in box, and its running Ice Cream Sandwich. Is there a way to load Ubuntu on it? Also the tablet is very buggy and crashes often, would that be due to the tablet or the OS? According to Theorem 8.8 in Time Series A.W. van der Vaart an ARMA process $$\phi (L)X_t=\theta(L)\epsilon_t$$ has a  unique stationary solution $X_t=\psi(L)\epsilon_t$ with $\psi=\theta/\phi$ if $\phi$ has no roots on the complex unit circle. This would imply that the explosive process, with $\rho&gt;1$, is a stationary process $$X_t=\rho X_{t-1}+\epsilon_t$$ with stationary solution $X_t=\sum_{i=1}^\infty \rho^{-i}\epsilon_{t+i}$.  Now indeed $\sum_{i=1}^{\infty} \rho^{-i} &lt; \infty$ so that weak stationarity can be proved by using this representation.   However, here on stackexchange I see a lot of question/answers that suggest that the process above is not stationary (see for example , ). In particular, the accepted answer of the latter question claims that the process is non-stationary by simulating a series and showing it displays explosive trending behaviour.  I think the only way to reconcile the theorem I mention above and the plots in the accepted answer of () is the following: the explosive process is indeed stationary but non-ergodic, that is, we cannot find the statistical properties of $X_t$ such as $\mathbb{E}(X_t)=\mu$ by observing a single infinitely long sample path of the explosive process, mathematically: $$\lim_{t \to \infty}\frac{1}{t}\sum_{t=1}X_t \neq\mathbb{E}X_t$$  Is this reading correct? Is there a way in javascript to detect if the unload event is caused via a refresh, the back button, or closing the browser? I am trying to include long URLs in a bibliography, but those often overflow into the margin of the document.  I have seen a number of questions about breaking URLs in bibliography, but after trying every solution, I have not found anything that completely solves my problem. There are apparently a number of ways to allow LaTeX to break line at different places in URLs. But in my case, even though LaTeX breaks line at some places, I still get overflow for some URLs.  Here is the code I am using:  \usepackage[hyphens]{url} \usepackage{hyperref} \hypersetup{breaklinks=true}  \urlstyle{same} \usepackage{cite}   and with these bibliography entries:  @misc(webrtc,     organization={W3C WebRTC Working Group},     title={WebRTC 1.0: Real-time Communication Between Browsers},     howpublished={\url{http://www.w3.org/TR/2012/WD-webrtc-20120821/}} )  @misc(android_javascriptinterface,     organization={Android Open Source project},     title={Android WebView addJavascriptInterface reference},     howpublished={\url{http://developer.android.com/reference/android/webkit/WebView.html#addJavascriptInterface(java.lang.Object, java.lang.String)}} )   I'm getting this, for example:     The entry about Android overflows in the border: my column should end where you can read "browsers." on the first line or "http:" on the third line.  Edit: when trying to create a minimal example for my problem, I noticed that this only occurs when using the IEICE template file. You can find it here: , but if you do not want to have to extract the archive and locate the correct file, I have put it there also:   Here is my minimal example:  .tex file:  \documentclass[paper]{ieice}  \usepackage[hyphens]{url} \usepackage[hidelinks]{hyperref} \hypersetup{breaklinks=true} \urlstyle{same} \usepackage{cite}  \title{Test document} \begin{document}  this\cite{upnp_spec} is a test\cite{android_javascriptinterface}  \bibliographystyle{plain} \bibliography{refs_min}  \end{document}   and refs_min.bib file:  @misc(android_javascriptinterface,     organization={Android Open Source project},     title={Android WebView addJavascriptInterface reference},     howpublished={\url{http://developer.android.com/reference/android/webkit/WebView.html#addJavascriptInterface(java.lang.Object, java.lang.String)}} )  @misc(upnp_spec,     organization = {UPnP Forum},     title = {UPnP Device Architecture 1.1},     howpublished = {\url{http://upnp.org/specs/arch/UPnP-arch-DeviceArchitecture-v1.1.pdf}} )   Is there any way I can fix this ? The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group, I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question. My Question is :   Why is the name Quotient Group kept,normally in the case of division, let us take the example of $\large \frac{16}{4}$ the Quotient of the Division is '$4$' which means that there are four '$4$'s in $16$, I mean we can find only $4$ elements with value $4$     So how can we apply the same logic in the case of Quotient Groups,like consider the Group $A$ and normal subgroup $B$ of $A$, So if $A/B$ refers to "Quotient group", then does it mean:           Are we finding how many copies of $B$ are present in $A$??, like in the case of normal division, or is it something different ??       I understood the Notion of Cosets and Quotient Groups,b ut I want a different Perspective to add Color to the concept. Can anyone tell me the necessity and background for the invention of Quotient Groups?  Note: I tried my level best in formatting and typing with proper protocol,if in case, any errors still persist, I beg everyone to explain the reason of their downvote (if any), so that I can rectify myself, Thank you. Suppose f is a twice differentiable function with $f(0)=0$ and $f(1)=1$, $f'(0)=f'(1)=0$. Then show that $|f''(x)| > 4$ for some $x$ in $(0,1)$ Man/Tree Symbiosis It was an old book... I'm pretty sure it was a book, and pre-1990's I think. The basic plot was this man somehow interacted with this tree, and as a result he became a carrier of the tree's 'seed'. They were in a symbiotic relationship: the man got a much longer life, more strength, etc.; but when he did die the tree would grow up from his body.