By numeric experiment, I have noticed that $$ \underset{i=0}{\overset{k}{\sum }}(-1)^{i+k} (i+n-1)^k \binom{k}{i} = k! $$ for any integer $n$ and $k \ge 1$. How can we prove this?
The successive difference of powers of integers leads to factorial of that power. Here's the formula:     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$   Can anyone give a proof of this result?  Note: The original question was to prove the more general     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$$   for an integer $l$, which some of the answers address.
What are the criteria for questions to be selected for Hot Network Questions? What are neutral density filters and how do I use them to create long exposures in daylight? I've seen a few related posts on the site but nothing specifically on this point:  I'd like to be able to take long exposure photos during daylight (like the examples in ). I'm told that you can do this by using a neutral density filter. So my question is: what is a neutral density filter exactly, and how is it used to achieve that smooth long exposure effect? When I call Response.Redirect(someUrl) I get the following HttpException:     Cannot redirect after HTTP headers have been sent.   Why do I get this? And how can I fix this issue? Load your bag or Load your bike? Probably, another debatable question from my side.  When carrying items while riding, like water bottle, repair kits, pumps, or other stuffs that you might need, What is the best option.... to carry them?   On your cycle at saddle bag, handlebar bags, or pannier bags or etc.  or, On backpacks, like hydrant pack, or simple travel pack.   While loading on cycle, it will increase the weight of the bike, thus affect a lot of things, like speed, traction, more reactive on corners. Like while using pannier bags.. it will increase rear weight, making it difficult on uphill.   Also while loading on backpack, it will tire you down fast if its a big load, decreases your flexibility, agility &amp; etc.  So what is right way to do it?  Update:  The type of ride I generally, go to are commuting, mountain biking, &amp; trailing. And the items I generally carry are   Commuting: Water bottle(x2), Laptop (17"), books, copies, repair kits, hand pump, few other accessories. Mountain Biking &amp; Trailing: Water bottle(x2), repair kits, extra tubes, hand pumps, candybars &amp; few other accessories Requesting raw data from previously published research Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to $2^{2048}$?  I understand this would take an immense amount of time to do, but it would only be done once. Obviously, computing such an index would take a considerable amount of time and would have to be done with someone having the right amount of resource. Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index? How long would that take in permutations and perhaps with the fastest computer in the world? There does not exist any integer $m$ such that $3n^2+3n+7=m^3$ I have this really hard problem that I am working on and I just don't seem to get it. The question is: let $n$ be a positive integer; prove that there does not exist any integer $m$ such that $3n^2+3n+7=m^3$.  Please explain in detail otherwise I might not understand. I am also open for any methods that might help. Thank you. Please help. Primary question: A common speaking mistake is to exchange syllables of words, saying "It's trace rhyme!" instead of saying "It's race time!", or pronouncing "kickin' chackatory" instead of "chicken cacciatore." Is there an existing name in poetry, linguistics, or English studies for switching syllables of (quasi-)consecutive words? I am not especially interested in the case when switching syllables results in interpretable homonyms or portmanteaus, like in the first example.  Secondary question: if not, what would you call it?  Tertiary question: is there somewhere a (playful?) taxonomy of fruitful or unfruitful lingual aberrations? I recently re-watched Contact for the first time since my childhood, and while I thoroughly enjoyed it, I realized that there was something I'd never understood: why couldn't they corroborate Ellie's account of her journey?  As far as I could tell, the pod fell harmlessly through the machine.  That is, it didn't break anything, and the machine wasn't destroyed.  I don't recall any dialogue about circuits being fried or the system shutting down, so presumably it wouldn't be hard to simply hoist the pod up and go again.  I imagine the energy draw was substantial on the local Japanese power grid, but given the amount of effort and money already invested in the machine, surely "Round 2" wasn't out of the question?  I mean, they almost certainly expected that this machine would get more than one good use out of it, anyway!  So given Ellie's vehement insistence that the machine had worked, the public nature of the controversy surrounding her testimony, and the government's (secret) findings that her device had recorded 18 hours of static, why wasn't it the obvious choice to have someone else suiting up the next day, ready to drop through the machine and corroborate her story? This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it? Bolzano-Weierstrass Theorem Proof Abbott (Ax. of Choice) In Abbott's Proof of the Bolzano-Weierstrass Theorem, does Abbott use any form of the Axiom of Choice ?   I think he is since he chooses an $a_{n_k} \in I_k$ where there are multiple such $a_{n_k}$. Let $X_1,...,X_n$ be mutually independent RVs. Suppose $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$.    How can I show that:  $$\sigma (X_1,...,X_n) \perp \mathcal F$$ ?  What I have tried:  $\sigma (X_1...X_n) = \sigma(\cup_{i=1} ^n \sigma(X_i))$ holds. I think I need some Dynkin's lemma-ish argument but can't see how. Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$ Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}.$ $f$ is a constant function? Flag as Broken Link I keep finding broken links in posts, currently I add a comment in the hope someone will return to update it or if I find a link I edit the answer.   I imagine this will only increase over time so I wonder if there could be a standard way to manage this?  My idea is adding a new flagging reason like "contains broken links", then it will go to a review queue where people will be able to have a look and fix the broken links. The cardinality of a language over a set of variables $V$ with $\#V= κ$ In predicate logic, to define a language, we consider the following $3$ sets:  (1) $V$, the set of variables;  (2) $F$, the sets of function symbols;  (3) $P$, the set of predicate symbols with their arities.   Now I am concidering about determine the cardinality of a language over a set with arbitary cardinality, $κ$(Note that $κ$ can be either finite, countablely infinite or uncountablely infinite ).  May I please ask:  If the cardinality of $V$, $\#V = κ$, what conditions on $V$ , $F$ and $P$ will give us a language over $V$, $F$ and $P$ with cardinality $κ$?  And may I please ask if it is the fact that if $\kappa$ is finite, then to get a language with cardinality $\kappa$ we need $(V+F+P)^{&lt;\omega}=(\kappa+F+P)^{&lt;\omega}=\kappa$, so we can never have a language of cardinality $\kappa$. I think it is something weird here. Am I wrong with something?  I had googled it but I have get nothing from websites, now I am confused. Could someone please help? Thanks so much!  EDIT: Latter I am told that in my course, the cardinality of a language refers to the set of expression instead of this of the set of strings. So now I need to consider further about when a string makes sense.  EDIT':Now I am considering that if we can find out a general way to determine the cardinality of expression, which may related to combinatorics. See this question:  Any help would be appreciate. My server is constantly being attacked I am fairly new to the system administration world. I've been working on an application recently and when I check my application server logs, I constantly get various IP addresses trying to ssh into my server by brute force. Here is an example of my server log:   Feb 14 04:07:20 foodwiz3 sshd[1264]: error: Could not load host key: /etc/ssh/ssh_host_ed25519_key Feb 14 04:07:21 foodwiz3 sshd[1264]: reverse mapping checking getaddrinfo for coenamor.columbiansabbatical.com [23.249.167.223] failed - POSSIBLE BREAK-IN ATTEMPT! Feb 14 04:07:21 foodwiz3 sshd[1264]: pam_unix(sshd:auth): authentication failure; logname= uid=0 euid=0 tty=ssh ruser= rhost=23.249.167.223  user=root Feb 14 04:07:23 foodwiz3 sshd[1264]: Failed password for root from 23.249.167.223 port 32997 ssh2 Feb 14 04:07:23 foodwiz3 sshd[1264]: Received disconnect from 23.249.167.223: 11: Bye Bye [preauth] Feb 14 04:13:04 foodwiz3 sshd[1289]: error: Could not load host key: /etc/ssh/ssh_host_ed25519_key Feb 14 04:13:05 foodwiz3 sshd[1289]: reverse mapping checking getaddrinfo for coenamor.columbiansabbatical.com [23.249.167.223] failed - POSSIBLE BREAK-IN ATTEMPT! Feb 14 04:13:05 foodwiz3 sshd[1289]: pam_unix(sshd:auth): authentication failure; logname= uid=0 euid=0 tty=ssh ruser= rhost=23.249.167.223  user=root Feb 14 04:13:07 foodwiz3 sshd[1289]: Failed password for root from 23.249.167.223 port 41562 ssh2   Is this something that is fairly normal, or should I be worried/doing something about it? How to pass the password to su/sudo/ssh without overriding the TTY? How to reinstall Windows 10 on new hard disk QGIS query syntax differs between attribute table and feature filter Using QGIS 2.16.2, I can successfully select records from a shapefile attribute table using the following expression:  left("start_time", 10) = '2015-08-01'   Please note that start_time is a string field.  However, when I copy and paste the same expression into the shapefile's Layer Properties>General>feature filter>Query Builder I get the following error:    I would expect that the same expression would work everywhere in QGIS.  What am I doing wrong? The plugin generated X characters of unexpected output during activation (WordPress)