Binomial Coefficients Summation $\sum_{k=0}^n a^k \binom{n}{k}^2$ Let $a&gt;0.$ Is there a closed form expression for the following summation: $\sum_{k=0}^n a^k \binom{n}{k}^2$?
Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$ I am trying to evaluate the finite sum \begin{equation} f(u)=\sum_{k=0}^n{n\choose k}^2u^k,\quad 0&lt;u\le1 \end{equation}  In an first attempt, I think of the generating function \begin{equation} (1+x)^n(u+x)^n = \sum_{k\ge0}{n\choose k}x^k\sum_{k\ge0}{n\choose k}u^kx^{n-k}=\cdots+x^n\sum_{k\ge0}{n\choose k}^2u^k+\cdots \end{equation}  which means that $f(u)$ is the coefficients of the term $x^n$.  Expanding $(1+x)^n(u+x)^n$ into $[1+(1+u)x+x^2]^n$, and using the multinomial expansion, I get an expression for the coefficient of $x^n$. However, such an expression is more complicated than $f(u)$. It seems I am making the problem even more difficult.  Can someone help me find the value of $f(u)$.  Thank you.
Is there such a thing as a switch that can be actuated automatically? I'm interested in a toggle switch that can be toggled without user input. That is, the physical state can be toggled electrically. It would need to incorporate some sort of motor or magnetic actuator. My google-fu is weak in this area, and I haven't been able to come up with the correct combination of search terms.  Note that I'm not talking about a relay; I'm talking about a normal toggle switch:    Except that it can be toggled without user input. As in, the lever physically moves. Like a  except that the mechanism is internal to the switch. "is a field and used like a type" error Experience using dropdowns in-line of sentence "mad lib" style Do you have experience implementing text inputs or drop-downs in-line with other text or forms? Is this approach any better than just creating a separate field/label?  In the example below the user has the option to indicate when to be reminded again (in one, two, or three days). When they are reminded of an assignment, they simply get an email or text message. Fatou's lemma and measurable sets I don't know how can I imply Fatou's lemma for any measurable sets $A_k$, that is, $$\lambda(\liminf A_k)\le\liminf\lambda(A_k).$$ How can I prove it? And is there any example in $\mathbb R$ of sequence of measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim\lambda(A_k)=1$, but $\liminf A_k=\varnothing$ ? Thanks for your help! What is a good book to learn number theory? How is the determinant related to the inverse of matrix? Is the empty set a group? I know that a set G with a binary operation $*$ is a group, if:   $a*b\in G$, for all $a, b \in G$. $*$ is associative:   $$(a*b)*c=a*(b*c) \\ \text{for all }a, b, c\in G.$$   An identity element $e \in G$ exists, such that   $$a*e = e*a = a\\ \text{for all }a\in G.$$   For all elements $a \in G$, there exists an $a^{-1} \in G$, such that:   $$a*a^{-1} = a^{-1}*a=e.$$  Can I use that to show that the empty set is a group? I accidentally killed a noble, now every guard wants to kill me. Is there any way I can get rid of the bounty? int t=-1'd1; //trying to assign -1 to t; bit vb[7:0]; // simple declaration  bit matrix[0:1][0:7]='{'{0,1,0,0,0,0,0,0},'{0,0,1,0,0,0,0,0},'{1,0,0,1,0,0,0,0},'{0,0,0,0,1,0,0,0},'{0,0,0,0,0,1,0,1},'{0,0,0,0,0,0,1,0},'{0,0,0,0,1,0,0,1},'{0,0,0,0,0,0,0,0}};// initialising multi-dimensional array.  I am using QuestaSim to compile and simulate. this is the report: # -- Compiling module test # ** Error: (vlog-13032) test.v(8): near &quot;=&quot;: Syntax error. # ** Error: (vlog-13032) test.v(9): near &quot;;&quot;: Syntax error.  I am somewhat New to the Verilog. Please help me in fixing these errors. How do I install applications in Ubuntu? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?   I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.  Thank you. Compute: $\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ where $A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)$ I have read answers to questions like  but I couldn't find a general rule for using "you" or "yourself", "him" or "himself" that could explain things I see in some examples.  For example, I think that one normally say (according to examples I see in ):   You bring something with you. You have something in you. You have something on you. You pressed her to you.   But on the other hand one would say:   You draw attention to yourself. He killed himself. You can try it out for yourself.   As a general rule, it seems that the reflexive pronoun is used when the object is the same as the subject of the verb. But for prepositional phrases, it seems more complicated. Do some prepositions (like "with", "in", "on") always come with the accusative form of the pronoun, and some other (like "for") don't? What about "to", that seems to be followed by both in different circumstances? What are the current travel restrictions on individuals entering the United States from regions affected by Covid-19? This community wiki explains the current restrictions by the United States of America on the entry of individuals who may pose a risk of transmission of the SARS-Cov2, the virus which causes the Covid-19 disease. Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$ Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon &gt;0$       $\epsilon$-chainable Definition      (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ a finite set $\{ x_0,x_1,...,x_n\}$ of points in X with $x_0 = a$, $x_n=b$ and $d(x_{n-1},x_n)&lt;\epsilon$, $\forall$ n      Clopen Definition      A set is clopen $iff$ its boundary is empty      Connected Metric Definition      Let $(X,d)$ be a metric space. It is connected $iff$ it has no proper subsets which are clopen      The following are equivalent:         $X$ is connected   The only subsets of $X$ that are clopen are $\emptyset$ and $X$   $\nexists$ non-empty, disjoint, open sets $A$ and $B$ such that $X=A\bigcup B$   $\nexists$ non-empty, disjoint, closed sets $A$ and $B$ such that $X=A\bigcup B$         Theorem      Let $(X,d)$ be a metric space, and let $S$ be any subset of $X$. Then $S$ is closed $iff$ $S^c$ is open.     So how would someone show this exactly?  Let the set $A$ be the set of all elements in $(X,d)$ that can be reached from $a\in X$ in a finite number of steps with length less than $\epsilon$.  Does this mean that there should be no isolated points? (an isolated point would be at distance infinity from any other point, or distance zero from itself, since $\epsilon &gt;0$ we don't want this?)  Knowing that $X$ is connected $\implies$the only subsets of X that are clopen are $\emptyset$ and $X$  Also, by construction of A;   A is the set of all $\epsilon$-chains between any element and $a$  And, so $A^c = \emptyset$ if $A$ was $\epsilon$-chainable (because A has to have a path from all elements to a, for this to be true, the complement wouldn't have anything in it)  Leaving only $A$ as the remaining clopen set  Does this make any sense? Or should an alternative be used, and how to formalize it.  Thanks. How is an empty set truly "empty"? In a  question, an answerer says:  an empty bag is a bag with nothing inside it.  Makes sense, but I'm reading a textbook right now that says:  The empty set has only one subset (namely, itself).  Which seems odd to me. If a set is empty, why would it contain a subset? Regardless of whether or not the subset is also an empty set, the &quot;empty set&quot; still contains a subset, so how is it really &quot;empty&quot;? To go back to the empty bag analogy, it just sounds like my text book is saying:  an empty bag is a bag with nothing but an empty bag inside it.  Which doesn't make sense because then the bag isn't empty. I can't wrap my head around this concept. When you add an image strip to the Video Sequence Editor, it stretches/shrinks your image to fit the full output frame.  How can I add an image at its original resolution?  For example, if I add a 60x60 pixel image, it should appear very small in the final 1080p render. do not optimize/remove html I would like to have a page with some html content in it that CK Editor does not seem to support, in particular: I want to have a little &lt;form tag with a couple of &lt;input fields. I can enter it in Switch to plain text editor mode and save from there and it will keep the elements but when I want to edit the page, it'll go back to the Switch to rich text editor and remove my &lt;form html items. How can i set CD Editor to not remove any html it may not be able to render?  Thank you! For my own understanding, I am interested in manually replicating the calculation of the standard errors of estimated coefficients as, for example, come with the output of the lm() function in R, but haven't been able to pin it down. What is the formula / implementation used? I have read that controlling FDR is less stringent than controlling FWER, such as in :     FDR controlling procedures exert a less stringent control over false discovery compared to familywise error rate (FWER) procedures (such as the Bonferroni correction). This increases power at the cost of increasing the rate of type I errors, i.e., rejecting the null hypothesis of no effect when it should be accepted.   But I was wondering how it is shown to be true mathematically?  Is there some relation between FDR and FWER?