Consider the number $b_h$ of binary trees of height $h$, where height is being measured by the number of levels. An empty tree has height $0$, a single node binary tree has height $1$, and there are three trees of height $2$. Find a recursive algebraic equation for a sequence $b$, and prove for all natural numbers $h$ that $b_{h}$ is the number of binary trees of height $h$.   Assuming $b_{1} = 1$ binary Tree.  Then $b_{2} =  3$ binary Trees.  The difference is increasing by $+1$, $+2$, ..., $+(h-1)$.  How can we get the recursive algebraic equation for this? I'm really confused.
I'm trying to figure out a recursive formula for the number of possible distinct binary trees at any depth d. I haven't been able to find any sort of sources on this.  basically, at depth 0, the only binary tree is the empty one so 0, 1  then at depth 1 the only possibility is a single node so 1, 1  at depth = 2 there are 3 possibilities, and at depth 3 I counted 21 possibilities.  I haven't been able to figure out some recursive reasoning for what 4 could be and so on. The only thing I have in place is that there will always be 1 for the full tree.
Proving “The sum of $n$ consecutive cubes is equal to the square of the sum of the first $n$ numbers.” This   states:      Example $\boldsymbol 3$. The sum of consecutive cubes. Prove this remarkable fact of arithmetic: $$1^3 +2^3 +3^3 +\ldots +n^3 =(1 +2 +3 +\ldots +n)^2.$$   “The sum of $n$ consecutive cubes is equal to the square of the sum of the first $n$ numbers.”      In other words, according to Example $1$: $$1^3 +2^3 +3^3 +\ldots +n^3 = \frac{n^2 (n+1)^2}{4}.$$   Should: $$1^3 +2^3 +3^3 +\ldots +n^3 = \frac{n^2 (n+1)^2}{4}$$ not be: $$1^3 +2^3 +3^3 +\ldots +n^3 = \frac{n^3 +(n + 1)^3}{2^3}$$ as everything in the left-hand side is cubed? Make display of reputation in revisions list consistent with the user card display The user card in the revision list, for example , shows the exact amount of reputation:  This is not consistent with how the flair works on (all?) other places in the site, e.g. under a post. There, the amount is displayed in units of thousands (k):  (or, in , millions (m).) It used to be consistent, see e.g. : "Do you want to continue[Y/n]?" Why the upper case? I am new to Ubuntu and some little thing start to bug me. Such as when trying to install a program, Ubuntu asks,     Do you want to continue[Y/n]?   What is bugging me is why the letter "Y" is upper case and "n" is lower case. Do they have any significant reason why one is upper and the latter is lower? Whatever case I enter it doesn't change the outcome: it either continues or not. Are they trying to notify us that we can enter both and it won't change anything? Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$ Show an Ideal is the principal ideal for some polynomial. Let $F$ be a ﬁeld and $R = F[X]$. Suppose $I$ is an ideal of $R$. Show that $I = (p(X))$ for some $p(X)$ in $F[X]$. (Hint: consider a polynomial $p(X)$ of least degree in $I$.)  I'm trying to do this question but can't seem to start. if anyone could give me a hint that would be appreciated.  My instinct was to take a monic polynomial (which exists since we are in a field) of degree $1$ and divide given the first two parts of the question were about polynomial division.  Thanks. In bash, calling foo would display any output from that command on the stdout.  Calling foo &gt; output would redirect any output from that command to the file specified (in this case 'output').  Is there a way to redirect output to a file and have it display on stdout? Big Greeks and commutation Why does Unix store timestamps in a signed integer? Why is a signed integer used to represent timestamps? There is a clearly defined start at 1970 that's represented as 0, so why would we need numbers before that? Are negative timestamps used anywhere? I use LINQ to Objects instructions on an ordered array.  Which operations shouldn't I do to be sure the order of the array is not changed? Prevent back button after logout Break a long command to multiple lines I got a shell script and inside I got the command as below #!/bin/sh sh ../jboss/bin/standalone.sh --server-config=standalone-full.xml -Djboss.node.name=node1 -b 0.0.0.0 -bmanagement 0.0.0.0  How can I break the line to multiple lines for better understanding and management #!/bin/sh sh ../jboss/bin/standalone.sh    --server-config=standalone-full.xml   -Djboss.node.name=node1   -b 0.0.0.0   -bmanagement 0.0.0.0  Tried with backslash at the end of each line and it doesnt work. "(" or "\left(" parentheses? What is the difference between ( and \left( in LaTeX? Sometimes, when the content is small, it does not seem to matter which pair I use.  What is the best practice when it comes to which parentheses use? I was so good with windows, now I’m feeling like a complete idiot.  I have an SSD for ubuntu and want to install the majority of my programs to a separate HDD but I have no clue how. All searches I have tried only turn up with dual-boot installations.   (RESOLVED -> ) I am also having trouble locating "/etc/". Specifically, to modify Conky. Searching for this term only turns up instances where people are telling you to access this file, not how to get to it. Poincaré's theorem about groups A problem from Isaacs' FGT This is problem 3B.12. from the afore mentioned book. It asks to show that, if $G$ is a solvable Frattini-free group (that is, $Φ(G)=1$) and $H$ is a subgroup of some maximal subgroup $M$ of $G$, then $G$ contains a subgroup of index $|M:H|$. It is easy to see that, if   $N \unlhd G$ and $M\cap N=1$, then $G=MN$, $M\cong G/N $ and $H\cong K/N $ for some $ K \leqslant G $, where $K$ contains $N$. Then $|G:K|=|G/N:K/N|=|M:H| $ and we are done. However, the hypothesis will not necessarily hold. But, if $N$ is minimal among normal subgroups of $G$ (in the sense of containment), then $N$ is elementary abelian, hence nilpotent, which implies that $M \cap N \lneq N_G(  M \cap N ) \cap N $. Thus, $M \lneq N_G(  M \cap N )$ and by the maximality of $M$ we deduce that $M \cap N \unlhd G$. So, for this particular $M$, if it is not true that there exists some normal subgroup of $G$ with which it intersects trivially, then $M \cap N = N$ by the minimality of $N$. Hence $M$ contains every minimal subgroup of $G$.     How should I proceed from here? If dark matter only interacts with gravity, why doesn't it all clump together in a single point? I'm a complete layperson. As I understand, dark matter theoretically only interacts with the gravitational force, and doesn't interact with the other three fundamental forces: weak nuclear force, strong nuclear force, and electromagnetism.  Those are my understandings going in. If I'm wrong, please correct me. I've done some googling, and I haven't found anything confirming or denying that dark matter is affected by either of the fundamental nuclear forces.  So since dark matter only interacts with gravity, what causes any dark matter particle to be repelled from another? If they can pass freely through each other, and they are gravitationaly attracted to each other, why don't such particles clump together in a single 'point' in space?   It seems to me that particles occupying a single 'space' are philosophically not distinct particles, but I don't know how actual physics would play into this.  Edit , author's credentials unknown, but implicitly claims to be a physicist or astronomer, says "...[P]hysicists generally take all dark matter to be composed of a single type of particle that essentially interacts only through gravity."  Edit 2 The author is this , "Professor of Science on the physics faculty of Harvard University." Best word wrap algorithm? Word wrap is one of the must-have features in a modern text editor.  How word wrap be handled? What is the best algorithm for word-wrap?  If text is several million lines, how can I make word-wrap very fast?  Why do I need the solution? Because my projects must draw text with various zoom level and simultaneously beautiful appearance.  The running environment is Windows Mobile devices. The maximum 600&nbsp;MHz speed with very small memory size.  How should I handle line information? Let's assume original data has three lines.  THIS IS LINE 1. THIS IS LINE 2. THIS IS LINE 3.   Afterwards, the break text will be shown like this:  THIS IS LINE 1. THIS IS LINE 2. THIS IS LINE 3.   Should I allocate three lines more? Or any other suggestions? ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­ When is it kosher to delete one's own answer? According to , the only restriction on self-deleting an answer is that you can't delete an answer that's been accepted.  But I've also heard of users getting in trouble for deleting (non-accepted) answers, and there's the CC-SA license to consider, so -- quite aside from what is technically possible -- my question is when it's permitted to delete your own answer.  Some cases to consider:   You now believe the answer is incorrect. Your answer is correct, but another answer covers the same territory better. You aren't satisfied with the quality/rigor of your answer, e.g. it's speculative and you didn't back it up (on a site that expects that). Something in the answer embarrasses you and you'd rather just make it go away. Your answer wasn't well-received by the community (whether that's "meh" or "make the rep bleeding stop please"). (I'm sure there are others; this isn't meant to be an exhaustive list.)   Does voting matter, either in absolute terms or in comparison to other answers?  For example, maybe it's ok to delete a lower-voted one but not the top-voted answer, or one with score > N??  There is a badge, Disciplined, for deleting your own post of score 3 or higher.  This tells me that it's sometimes ok to delete upvoted answers.  But I'm not sure how much policy we can infer from badges.  Could we have some clarification please? When to use virtual destructors? I have a solid understanding of most OOP theory but the one thing that confuses me a lot is virtual destructors. I thought that the destructor always gets called no matter what and for every object in the chain. When are you meant to make them virtual and why? Centos mysql version is 5.5 however PHPmyadmin still says version 5.1 When i run the following in my console:  [root@****~]# mysql -u root -p -e 'SELECT VERSION();' Enter password: +-----------+ | VERSION() | +-----------+ | 5.5.39    | +-----------+   Which should be the correct version.  However when i enter my PHPMYADMIN on my server it has the following specs:  Server: Localhost via UNIX socket Program: MySQL Programversion: 5.1.73 - Source distribution   So which version is the correct one and how do i make sure that the database is running on 5.5?  Note  I have already restarted sql several times without any changes