Please share if you have a faster approach, but I’ll share what I have so far. I think it reduces to some combinatorial problem, I’d like to see a combinatorial argument if possible.   From computing for small $n$ in wolfram alpha, I think the answer is \begin{align*} \ln(x^2-x+1)&amp;=\sum_{n=1}^\infty (-1)^n \frac{a_n}{n} x^n \\ &amp;= -\frac{1}{1}x^1+\frac{1}{2}x^2-\frac{2}{3}x^3+\frac{1}{4}x^4-\frac{1}{5}x^5+\dots \end{align*} where $a_n=2$ if $n=3,9,27,81,\dots$ and $a_n=1$ otherwise.  We know $$ \ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots $$  So $$ \ln(1+(x^2-x))=\frac{x}{1}(x-1)-\frac{x^2}{2}(x -1)^2+\frac{x^3}{3}(x-1)^3-\dots$$  Now we compute the coefficient of $x^n$. \begin{align*} x^8 \to -\frac{1}{4}\binom{4}{4}+\frac{1}{5}\binom{5}{3}-\frac{1}{6}\binom{6}{2}+\frac{1}{7}\binom{7}{1}-\frac{1}{8}\binom{8}{8}=\frac{1}{8}\cdot8! \\ x^9 \to \frac{1}{5}\binom{5}{4}-\frac{1}{6}+\frac{1}{7}\binom{7}{2}-\frac{1}{8}\binom{8}{1}+\frac{1}{9}\binom{9}{0}=-\frac{2}{9}\cdot 9! \\ x^{10} \to \frac{1}{5}\binom{5}{5}-\frac{1}{6}\binom{6}{4}+\frac{1}{7}\binom{7}{3}-\frac{1}{8}\binom{8}{2}+\frac{1}{9}\binom{9}{1}-\frac{1}{10}\binom{10}{0}=\frac{1}{10} \cdot 10! \end{align*} In other words, I want this identity: $$ \sum_{r+k=n} (-1)^{r+1} \frac{1}{r} \binom{r}{k} = (-1)^n \frac{a_n}{n} n! $$  with $a_n$ defined as above. Any help to prove the identity would be appreciated. Any other faster approaches would also be appreciated. Thanks!  Also, this LaTeX was a pain to type. Is there a way to reduce it? (Like \frac to \f, \binom to \b)
Is there a closed form expression for $\text{n}^{\text{th}}$ coefficient of Maclaurin series of  $$\ln(1-x+x^2)$$
Problem with camera orientation in motion tracking sorry for my English, I write from Italy. I'll try to explain my problem. I'm tracking a footage with a solver error 0.2973. When I try to set the floor, the orientation of the camera is wrong (see screenshot).  So I tried to manually orient the camera, but even if well oriented respect to the xyz axes, geometry is incorrect as you can see from the screenshots below (the selected marker appears to be on the same plane but it is lower).  The video was recorded by a roof of a house so much higher than the road. It seems that Blender is not able to recognize this height and set the floor in a wrong point of view. Can someone help me solve this problem? Thank you Root filling up although it should have some space This has been perplexing me for some time. I'm on Ubuntu 11.04. My root partition is filling up and I can't figure out what's causing it. Even stranger, when I run df -h, this is the output:  Filesystem            Size  Used Avail Use% Mounted on /dev/sda2             9.2G  8.8G     0 100% /   As you see, there should be about 400 MB free, but Avail shows 0. What could be causing this? I've even uninstalled some programs to free the space and it fills up again. Suggestions on how to find out which, if any, files are filling up the space are also welcome. Stack Exchange app iPhone X optimization The Stack Exchange app doesn’t look very good on Apple’s brand new flagship phone yet.    Also the app can’t identify the device model either. (Says Unknown (iPhone 10,3))   App Version: 1.6.6.2 Device: iPhone X (iPhone10,3) OS Version: Version 11.1 (Build 15B93) Marshal "char *" in C# Given the following C function in a DLL:  char * GetDir(char* path );   How would you P/Invoke this function into C# and marshal the char * properly. .NET seems to know how to do LPCTSTR but when I can't figure out any marshaling that doesn't cause a NotSupportedException to fire when calling this function. What is the optimal path between $2$ fixed points around an invisible obstructing wall? I made a skin by using Skincraft and I exported it to minecraft.net. I chose my skin and uploaded it. At the top it says,      "Your skin has been changed! Please wait a few seconds for it to   update."   I went to Minecraft, clicked singleplayer and chose a random world at the top. I looked at my character, but it's still default. Multiplayer won't load for some reason. I don't know what I've done wrong! Did the Big Bang happen at a point? TV documentaries invariably show the Big Bang as an exploding ball of fire expanding outwards. Did the Big Bang really explode outwards from a point like this? If not, what did happen? I am writing my thesis and using latex for first time. I would like to modify the the first page of my thesis, as it shows only name of thesis, my name etc. I would like to move all  text little-bit to right-side, just on the first page, without effecting the other pages of thesis. Any idea how can I do it?  \documentclass[a4paper,      12pt,      twoside,      openright,      cleardoublepage=empty,      numbers=noenddot,      appendixprefix,      BCOR1.5cm,      bibliography=totoc      ]{scrbook}   \thispagestyle{empty} \begin{titlepage} \begin{center}  %xyz  \end{center} \end{titlepage} \clearpage In Book 1, Artemis et al escaped from the time stop by drugging themselves to sleep.  In the same time stop, Julius Root "accidentally" plugged Briar Cudgeon with a tranquilizer dart.  Cudgeon fell asleep, but did not disappear from the time stop.  Why not? What is the name of the tactic that politicians use to bury people with torrent of words? Some people write bloated books and long essays with skilful use of hooks, e.g. Jared Diamond; some others speak in long-drawn sentences with torrents of words, e.g. Noam Chomsky. It reminds me of a tactic politicians use in public discourse to waste people's time. I came across this word before but I can't retrieve it from either memory or the internet. How can I generate isometric sprite sheets from 3D animations? I've read some great on-topic example questions on Area 51.  My questions:   Should a proposed site's "Great On-Topic Example" questions be automatically included in its beta? Will a proposed site's "Great On-Topic Example" questions be automatically included in its beta? How to *quickly* get a list of files that have bad sectors/blocks/clusters/whatever? My wife's Windows 7 (64-bit) box has suddenly developed a SMART "disk is bad" status.  I'm attempting to copy everything off (no admonishments about lacking a backup regimen, please, I know already  :(  ) by creating a System Image across the network to a different machine, but it gets to a certain point and starts taking forever.  Doing a chkdsk reveals that certain files cause this by having many bad blocks (like dozens of thousands in a row, if the event log is any indication) and causing the system to do its standard try-to-recover-and-relocate-upon-access thing.  But this is taking so long, I'm afraid the disk will fail completely before I can get the damned thing copied.  However, several of the files so far have been ones that she has copies of elsewhere, so I am able to just delete them prior to retrying the backup to speed things up considerably.  So:  is there some tool or procedure that will try reading each file, and upon hitting a bad block, just tell me about it and skip to the next file?  So I can see which ones I can just dump and which I need to let it try to recover? How do I allow HTTPS for Apache on localhost? Can the definite article be omitted when referring to a person qualified by a noun? How can I measure the execution time of a terminal process? I'm trying to measure the execution time of a process that I call via the command line (i.e., I want to find out how long it takes to for the process to finish). Is there any command that I can add to the command calling the process that will achieve this? How do I convert between big-endian and little-endian values in C++? For clarity, I have to translate binary data (double-precision floating point values and 32-bit and 64-bit integers) from one CPU architecture to another.  This doesn't involve networking, so ntoh() and similar functions won't work here.  Note: The answer I accepted applies directly to compilers I'm targeting (which is why I chose it).  However, there are other very good, more portable answers here. Taylor expansion for $\arcsin^2{x}$ I stumbled upon this particular expansion that was included in  post.    $$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$    This caught my eye because I remember trying to derive a Taylor series for $\arcsin^{2}(x)$ a while ago without much success.  Can anyone prove this or point me to a material that would show a proof of this identity?  EDIT :  Feel free to use any mathematical apparatus at hand. I'm not interested in a proof fit for a certain level, nor am I looking for utmost elegance (though that would be lovely). What is wrong with this fake proof $e^i = 1$? $$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$  Obviously, one of my algebraic manipulations is not valid. Auto revision summary has wrong plural: "added 1 characters in body" As seen at:           Update: I'm 101% sure the down votes were due to the lack of free hands.    Update 2: likes me, like me not, like me, likes me not... yeah !