# Tagged Questions

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### Find $a_1$ given that $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$

If $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$, then $a_1$ is .. The options are $1$, $2$, $99$ or $100$. I'm sure the problem is trivial, but I just don't understand what is meant.
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### Evaluate $\sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ where $n,k$ are fixed

Is there a general way/technique to evaluate $\sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ in terms of $r$, where we consider $n$ and $k$ fixed natural numbers and $n > k$? (here, ...
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### Consecutive terms in Pascal's Triangle

is it known whether or not there are infinitely many pairs of consecutive terms in this sequence: http://oeis.org/A006987 ? The sequence is the list of numbers expressible in the form ...
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### Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$

I am trying to evaluate the finite sum $$f(u)=\sum_{k=0}^n{n\choose k}^2u^k,\quad 0<u\le1$$ In an first attempt, I think of the generating function ...
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### Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
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### Given the sequence $3, 4, 11, 16, 42\ldots$ how can I derive a general formula for it?

Given a sequence $3, 4, 11, 16, 42\ldots$ how can I derive a general formula for this sequence? Is there any optimised approach? My approach: the given series is equal to summation of $\binom{n}{k}$ ...
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### Asymptotic behavior of $\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k$

I am looking to show that $$\lim_{n \rightarrow \infty}\frac{1}{e^n}\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k = 0.$$ In my application, $c = (e+1)/2 \approx 1.85914\ldots$. I have been ...
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### Infinite Sum with Combination

I am trying to figure out what the following sum converges to: $$\sum_{n=0}^\infty {6+n\choose n}x^n(6+n),\qquad\qquad0<x<1$$ An answer would be great, but if you have an explanation, that'd ...
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### Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$ My work: $\sqrt8=\bigg(1-\dfrac12\bigg)^{-\frac32}$ Now, I suppose there is some "binomial ...
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### Upper bound for $\sum_{j=0}^i {i \choose j}^{n}$

Is there an upper bound for sums of powers of binomial coefficients? I have $$\sum_{j=0}^i {i \choose j}^{n}$$ where $n$ is a positive integer. I am hoping this will help me solve Limit for a ...
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### Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$

I'm trying to calculate the limit for the sum of binomial coefficients: $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$
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### Validating a proposition

Proposition: For all $k,n\in\mathbb{Z^+}$ $s.t$ $n\lt4$ $2{n\choose n}+{n\choose n-1}+...+{n\choose k-(n-2)}=2^n$ for $1\le k\le n-1.$ I understand that this proposition is invalid, so are there ...
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### Upper bound for sum

I am trying to get an upper bound the following sum: $$S_{n,r}=\sum_{i=0}^n \binom{n}{i} \left(\frac{\binom{n}{i}}{2^n}\right)^{r} .$$ Any hints would be greatly appreciated. I thought of using ...
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### Express number of ways integer can be written as coefficient in generating series

Question: "Express the number of ways that an integer $n$ can be written as a sum of a cube of an integer $s\ge-1$ plus the fourth power of an integer $t$ plus the square of an odd integer $r$ as a ...
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### Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
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### Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$\prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n$$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
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### Does $\sum_{k=0}^n {n \choose k} x^{k^2}$ have a closed form?

Does $\sum_{k=0}^n {n \choose k} x^{k^2}$ have a closed form, similar to $\sum_{k=0}^n {n \choose k} x^{k} = (1+x)^n$?
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