Is it possible to find a closed form for the expression below? $\sum_{k=0}^n\binom{2n}{n} \binom{2n-2k}{n-k}$ I think it is possible to use a former solved one below. $\frac{1}{\sqrt{1-4x}} =  \sum_{n=0}^\infty \binom{2n}{n} x^n $
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.
Failed to save undefined: No COMPONENT named markup://svg found : Let $f(x) = x^2 \sin{\frac{1}{x}}$ for $x\neq 0$ and $f(0) =0$. (a) Use the basic properties of the derivative, and the Chain Rule to show that $f$ is differentiable at each $a\neq 0$ and calculate $f'(a)$. You may use without proof that $\sin$ is differentiable and that $\sin' =\cos$.  Not even sure what this is asking.  (b) Show that $f$ is differentiable at $0$ and that $f'(0) =0$.  $\frac {f(x)-f(0)}{x-0} \to \lim_{x \to 0} x \sin(1/x)$. $x \sin(1/x) \leq |x|$ and $\lim_{x \to 0} |x|=0$. Thus $f(x)$ is differentiable at $0$; moreover $f^{'}(0)=0$.  (c) Show that $f'$ is not continuous at $0$.  $f{'}(x)=x^{2} \cos(1/x) (-x^{-2}) + 2x \sin (1/x)$. In pieces: $\lim_{x \to 0} \cos (1/x)$. $f^{'}(0-)$ nor $f{'}(0+)$ exists as $x \to 0$ $f^{'}(x)$ oscillates infinity between $-1$ and $1$ with ever increase frequency as $x \rightarrow 0$ for any $p&gt;0$ $[-p,0]$, $[-p,p]$ or $[0,p]$ $f$ is not continuous. Question: How to show more rigorously? Is "do something by date X" inclusive? Possible Duplicate:          For example, if John says:      Return it to me by March 24th.    Does it mean that I need to return it to him before 23:59 p.m. March 24th, or before 23:59&nbsp;p.m. March&nbsp;23rd? I met this equation frequently in Guass-Newton optimizations. But I dont understand why the left and right side of the equation can be equal.  Lets say the Jacobian is $2$ by $2$   and Hessian is $$\begin{bmatrix}\frac{\partial^2f_1}{\partial^2 x_1 } &amp; \frac{\partial^2f_1}{\partial^2 x_2  } \\ \frac{\partial^2f_2}{\partial^2 x_1 } &amp; \frac{\partial^2f_2}{\partial^2 x_2  }\end{bmatrix}$$   But the right hand side of the equation is  $$J^TJ =\begin{bmatrix} \left( \frac{\partial f_1}{\partial x_1} \right)^2 + \left( \frac{\partial f_2}{\partial x_1} \right)^2  &amp;  \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1}\\ \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1} &amp;\left( \frac{\partial f_1}{\partial x_2} \right)^2 + \left( \frac{\partial f_2}{\partial x_2} \right)^2\end{bmatrix} $$  Why can these two be equal as presented in papers? Is there an alternative to the word "coincidence" to describe when two say the same word spontaneously? Is there a word to describe a scenario where two people having a conversation utter the same word/phrase together, simultaneously, and unconsciously? Something else than just a coincidence. I have a function with the same name, but with different signature in a base and derived classes. When I am trying to use the base class's function in another class that inherits from the derived, I receive an error. See the following code:  class A {     public:     void foo(string s){}; };  class B : public A {     public:     int foo(int i){}; };  class C : public B {     public:     void bar()     {         string s;         foo(s);     } };   I receive the following error from the gcc compiler:  In member function `void C::bar()': no matching function for call to `C::foo(std::string&amp;)' candidates are: int B::foo(int)   If I remove int foo(int i){}; from class B, or if I rename it from foo1, everything works fine.   What's the problem with this? What is the best time to ask questions? How do I programmatically determine operating system in Java? Too big of a sofa Using residue theorem to integrate from $-\infty$ to $\infty$ I'm trying to integrate $$\int_{-\infty}^{\infty} {x^2 \over {(x^2 + 1)}^2(x^2 + 2x + 2)} $$ given that the function  $$f(z) = {z^2 \over {(z^2 + 1)}^2(z^2+2z+2)} $$ has residues $${9i - 12 \over 100},{3 - 4i \over 25}$$ at the poles $i$ and $-1+i$ respectively. From my understanding of this I have added the two residues (by the residue theorem, and because their respective poles lie in the upper half plane) and multiplied by $2\pi i$ and got an answer of $14\pi \over 100$  Have I done this right? Why is it bad to log in as root? I've often come across posts on forums or other websites where you see people joking in such a manner about running/logging in as root as if it's something awful and everyone ought to know about it. However, there isn't much that a search reveals on the matter.   It may be widely known to Linux experts, but I really don't know why. I remember always running as root when I first tried Linux years ago (Redhat and Mandrake) and don't remember running into any problems because of that.   There are actually some distros that have a bright red background with alert signs all over it as wallpaper for the root user (SuSe?). I still use the "Administrator" account for regular use on my Windows installation and haven't ever run into any problems there either. Tutorial on how to back up my full Ubuntu desktop and files? I would like to be able to back up all my settings and files to a dedicated external hard drive in a way that makes (1) all the backup files accessible and (2) would allow me to restore my system is there is a crash.   Is there an app for this or any pointers?  Thank so much. Calculate expected values using logistic regression coefficients The outcome of the logistic model is binary (0;1) - intervention that occurs in hospital (3838 interventions in 51096 cases). I am modeling the effect of age groups (60 to 74; 75 to 89 and 90+) and sex (female) on the outcome. My reference age group is male aged 18-59.  My logistic regression coefficients look like this:  intercept  -4.313 female      0.5117 age 60-74   0.4081 age 75-89   1.1761 age 90+     1.5612.   Corresponding odds ratios:  female      1.668 years60_74  1.504 years75_90  3.242 years90     4.764   Now I am trying to obtain a probability of an outcome = 1 for each case. What is the right way to do this? Messed up Blender models when exporting to Unity3D I have modeled a simple pocket watch in Blender. Every time I import it into Unity3D, regardless of whether I export it from Blender as an FBX or 3ds file, it always comes out missing faces! Why is this happening? I have never had problems importing models into Unity3D like this before.  Blender Screenshot    Unity Screenshots     It seems to me that different faces appear when I rotate it differently. How can I prevent this? Thanks! What is the best indicator in finding out whether a database is utilized or accessed?  I tried index statistics and  based on I/O utilization but none seem to be a good indicator.   Problem: I have a lot of databases on a development environment and due to space issues we are trying to decommission those that are not used without actually messing with the ones that are really needed. Is there a software utility to adjust screen gamma/brightness/contrast? I use to work at night and it is eye-hurting to watch at lightbulb-bright screens (I've got two - laptop's built-in panel and an external 18-inch CRT) in a dark environment. So I adjust my screens to be darker.  Can I do this a software way instead of pressing monitors' buttons that many times every time? By looking at this picture:    The earth is near the center of the universe. I've read that the universe look the same no matter where the observer is located. It is the same distance everywhere.  So I understand that for general relativity the universe need to be homogeneous and isotropic, so it will look the same no matter where I am.  But what if I'm on one of the planets near the right or left of the image, then if I draw the same picture of the universe, but from my perspective, then I would be also located in the center? If that's not the case (I'm actually near an edge), then part of my sky would be completely dark, and all the sky that way won't be isotropic? Is it possible to annotate equations (i.e. arrow signs/curly braces) to define each variable? I have quite a simple question. I was wondering if it was possible to do something like this in LaTeX:    But formatted a little more nicely.  It doesn't need to be arrows necessarily. Just some form of annotation marker. jQuery Event Keypress: Which key was pressed? With jQuery, how do I find out which key was pressed when I bind to the keypress event?  $('#searchbox input').bind('keypress', function(e) {});   I want to trigger a submit when ENTER is pressed.  [Update]  Even though I found the (or better: one) answer myself, there seems to be some room for variation ;)  Is there a difference between keyCode and which - especially if I'm just looking for ENTER, which will never be a unicode key?  Do some browsers provide one property and others provide the other one? Checking if array is multidimensional or not? What is the most efficient way to check if an array is a flat array of primitive values or if it is a multidimensional array? Is there any way to do this without actually looping through an array and running is_array() on each of its elements?