# Tagged Questions

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### How many numbers are less than million such that their digits sum is $\le 19$?

How many numbers are less than million such that their digits sum is $\le 19$? This question is a Generating-Functions exercise. The solution claims the answer is the coefficient of $x^{19}$ ...
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### Why is this function generate $a_n = 2^n(n+1)$?

Let $G(x) = \frac{1}{(1-x)^2}$ which generates the sequence $a_n = n+1$ How can one infer that $G(2x) = \frac{1}{(1-2x)^2}$ generates $a_n = 2^n(n+1)$? Thanks.
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### does this sequence of functions converges uniformly to Dirichlet function?

Let $r_{1},r_{2},...$ a sequence that includes all rational numbers in $[0,1]$. Define $$f_n(x)=\begin{cases}1&\text{if }x=r_{1},r_{2},...r_{n}\\0&\text{otherwise}\end{cases}$$ this sequence ...
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### How is this step completed?

User Did, did this step in his answer to my previous question: $$\sum_{k=0}^n{n\choose k}(zp)^kq^{n-k}=(q+pz)^n.$$ How is it done? Is it simply an identity, or something more?
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