I have two series, say $\{a_k\}$ and $\{b_k\}$ for $k=0, 1, \ldots$.  They are related to each other as  $$\sum_{r=0}^k \binom{k}{r} b_r = a_k.$$  I want to compute $b_k$'s.  Of course, I can do this using the following recurrence relation  $$b_k = a_k-\sum_{r=0}^{k-1}\binom{k}{r} b_r.$$  But this is quite time consuming as $k$ increases because the number of operations is $O(k)$.  Is there a possibility that we can have a recurrence relation to obtain $b_k$ in $O(1)$ time?
Suppose that sequences of real numbers satisfy:  \begin{align*}   a_n &amp;= \sum\limits_{r=0}^n \binom{n}{r} b_r \\ \end{align*}  Prove that:  \begin{align*}   (-1)^n b_n &amp;= \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s \\ \end{align*}  My work:  The $n=0$ case:  \begin{align*}   a_0 &amp;= b_0 \\   b_0 &amp;= a_0 \\ \end{align*}  Then $n=1$ case:  \begin{align*}   a_1 &amp;= \binom{1}{0} b_0 + \binom{1}{1} b_1 = b_0 + b_1 \\   -b_1 &amp;= \binom{1}{0} a_0 - \binom{1}{1} a_1 = a_0 - a_1 \\   b_1 &amp;= a_1 - a_0 = b_0 + b_1 - a_0 = b_1 \\ \end{align*}  The inductive step:  \begin{align*}   (-1)^{n+1} b_{n+1} &amp;= \sum\limits_{s=0}^{n+1} \binom{n+1}{s} (-1)^s a_s \\ \end{align*}  I'm not sure where to go from here. I tried substituting $a_s$ into that last equation and it didn't help. Any ideas?
Is there a terminal command that will log you out of your current desktop session and take you back to the login screen? How does item level and rarity affect the possible attributes? I have a friend who has got a computer that is not connected to the Internet. Is there any way to install software offline easily? Use of “f ” instead of “s” in historic, printed English documents I was at a museum in London yesterday, and one of the items on exhibit is a document from the eighteenth century. It uses the letter f a lot where s should be used—for example, in Majefty.     Did the letter f once have the same sound as s today, or were some of those words actually spoken with an f ? What does unwrapping a model do and why is it important? How to make shortcuts on Ubuntu 17.10? String vs. StringBuilder I understand the difference between String and StringBuilder (StringBuilder being mutable) but is there a large performance difference between the two?   The program I’m working on has a lot of case driven string appends (500+). Is using StringBuilder a better choice? Is black a color or absence of color? Java has a convenient split method:  String str = "The quick brown fox"; String[] results = str.split(" ");   Is there an easy way to do this in C++? "As a beginner, I am happy that I learned a new thing."  You will recognize this as your DoReMi-s.( at 4:52)  What is the meaning of the "AS" there and when do we use that?  In depth answers would be appreciated.  Similarly:     Do I have your word as a lawman not to shoot me down like a dog in the street?    Can you explain this sentence?   Can I understand it as a lawman wouldn't shoot down people on the street or wouldn't have his word or what? Disk space not freed up after deleting files and emptying Trash I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane.   There are several possible approaches but I'm not sure which is the best - both looks quite difficult to solve algebraically.   using polar equation relative to focus with $(R_1,\phi_1),(R_2,\phi_2)$, being coordinates of points $$ R_1 = \frac{a(1-e^2)}{1-ecos(\phi_1-\theta)} $$ $$ R_2 = \frac{a(1-e^2)}{1-ecos(\phi_2-\theta)} $$ then for given $\theta$ solve for semimajor axis $a$ and eccentricity $e$ using deffinition of elipse as a set of points of the same distance from both foci. Given 2 points of cartesian coordinates $(x_1,y_1),(x_2,y_2)$ and one focus in origin $(0,0)$. For each given distance parameter $L$ solve for coordinates of second focus $(x_f,y_f)$, $$ L = \sqrt{x_1^2 + y_2^2} + \sqrt{(x_1 - x_f)^2 + (y_1 - y_f)^2} $$ $$ L = \sqrt{x_2^2 + y_2^2} + \sqrt{(x_2 - x_f)^2 + (y_2 - y_f)^2} $$ I can also first rotate the coordinate system (or my input points) by given angle (which is my arbitrary parameter) and then use some simplified equation of ellipse which has major axis parallel to x-axis which has just 2 degrees of freedom. But even after this rotation I don't see much simplification of algebraic solution.    Or is there any better way?    The resulting equations are quite difficult to solve. I wonder if there is some trick to simplify it and make it more elegant.  Also, because I would later to implement it into computational code where it would be computed manytimes per second, I would prefer solution inf form which is fast to evaluate numerically ( prefferably without goniometric or any transcedental functions )    Why I'm interested ? - I need it for some optimalization of orbital transfers for spaceship orbiting the Sun. I asked also in physics.stackexchange, and people recomand me that I should rather go to math section. Reference for $\text{Var}(X)\le (b-a)^2/4$ I am not a statistician, but am working a proof for the upper bound of an expression which contains the variance of a variable which obtains its values from a closed interval, [0,1]. I have seen in several places (mostly forums and web pages) the result that $\text{Var}(X)\le \frac{(b-a)^2}{4}$ for $a\le X\le b$. I would like to use, and cite, this result but am having an unusually difficult time finding a solid reference. Could someone here recommend one for me? How to specify password in ssh command From the terminal I type: ssh user@ip and then it prompts for a password.  Is there a way to specify the password in the ssh command itself? I’m going to Germany for a semester exchange and I have a long-stay visa from Germany. Can I travel to the other Schengen states during this period with this visa or do I have to apply for a separate short stay Schengen visa?  My visa is German national visa type D. It is for a duration of 4 months, i.e., longer than 90 days. It says on the visa that the long stay visa is valid in Deutschland, which is different from my earlier visa, which I had when I visited Germany earlier; that one was a short stay visa and said it was valid in Schengen Staten.  Perhaps my current visa means that I can stay in Germany for a period of longer that 90 days can visit the other Schengen states if I don’t exceed the 90 day period. Solve recurrence relation $a_n = 3 a_{n - 3} + a_{n-1} + 2 n$ I am not able to solve the above recurrence relation which involves $n$. The initial conditions $a_0 = 1, a_1 = 2, a_2 = 3$, for $n \geq 0$. Please try to give closed form in terms of arbitrary $a_i \geq 1 $ for $i \geq 0$ I am trying to copy the contents of a folder to another folder in a different directory using terminal.  Would somebody be able to provide me an example of the command line syntax required to achieve this? Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence? Deriving Sum of a Geometric Progression Total combinations from number of unique combination? How can you find count of the total combinations possible if you have the count of unique combinations? Suppose I need to form a sum of $3$ using $\{1,2\}$ Unique combn. $(1+1+1)$, $(1+2)$ hence '$2$' unique but total is '$3$' i.e, $(1+1+1)$, $(1+2)$, $(2+1)$ I know the elements used to make the combination Note: Number of unique combinations (ie. 2 here) is known. How can I find the total number of combination given the unique combinations?