Question

Explain the last line of this proof (it's an identity I can't figure out)?

score 93 · Answer 1 · 2016-02-14 05:27:37

93

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Explain the last line of this proof (it's an identity I can't figure out)?

Published on 14 Feb 2016 - 5:27

score 494 · Answer 2 · 2026-03-26 17:33:39

494

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If $(1+2x)(1+x+x^2)^{n}=\sum_{r=0}^{2n+1} a_rx^r $ then find the required value

Published on 26 Mar 2026 - 17:33

score 773 · Answer 3 · 2016-02-15 12:33:43

773

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Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$

Published on 15 Feb 2016 - 12:33

score 85 · Answer 4 · 2016-02-16 02:19:03

85

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$(1+x)^{20}=\sum_{r=1}^{20}a_rx^r$ where $a_r=\binom{20}{r}$, find the value $\mathop{\sum\sum}_{0\le i<j\le 20}(a_i-a_j)^2$

Published on 16 Feb 2016 - 2:19

score 5.9k · Answer 5 · 2016-02-18 02:18:10

5.9k

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Binomial Theorem and Complex Numbers

Published on 18 Feb 2016 - 2:18

score 55 · Answer 6 · 2026-04-01 23:10:51

55

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Showing that percentage of the coefficients of $(x+y)^n$ being even tends towards $100\%$ when taking the limit of $n$ to infinity

Published on 01 Apr 2026 - 23:10

score 1.2k · Answer 7 · 2016-02-20 01:46:27

1.2k

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Binomial theorem alternating sum

Published on 20 Feb 2016 - 1:46

score 33 · Answer 8 · 2016-02-20 23:22:04

33

Views

How do I find all the coefficients of $x^0$ in this: $(x-\frac{2}{\sqrt x})^8$

Published on 20 Feb 2016 - 23:22

score 68 · Answer 9 · 2016-02-20 23:33:14

68

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Compute $\binom n0 + 2\binom n1 + \binom n2 + 2\binom n3+...$

Published on 20 Feb 2016 - 23:33

score 192 · Answer 10 · 2016-02-21 00:03:58

192

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How do I prove $2 \binom n2 + 4 \binom n4 + 6 \binom n6 + \cdots = n 2^{n-2}$

Published on 21 Feb 2016 - 0:03

score 3k · Answer 11 · 2016-02-22 11:50:25

3k

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Find the sum of the coefficients in the expansion of the given expression using Binomial Theorem.

Published on 22 Feb 2016 - 11:50

score 31 · Answer 12 · 2026-03-29 17:31:46

31

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Anti-symmetric ways

Published on 29 Mar 2026 - 17:31

score 1.5k · Answer 13 · 2016-02-22 21:53:19

1.5k

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Sum of the coefficients of polynomial $f(x)= (3x-2)^{107} (x+1)^4$

Published on 22 Feb 2016 - 21:53

score 71 · Answer 14 · 2016-02-22 23:19:16

71

Views

Combinatorial proof (catching fishes and eating some from them)

Published on 22 Feb 2016 - 23:19

score 114 · Answer 15 · 2016-02-23 02:38:00

114

Views

General strategies for evaluating sums in probabilistic/combinatoric problems: $\sum_n\sum_m {n+m \choose n}p^{n+m}(1-p)^{n+m}$

Published on 23 Feb 2016 - 2:38

List Question

Explain the last line of this proof (it's an identity I can't figure out)?

If $(1+2x)(1+x+x^2)^{n}=\sum_{r=0}^{2n+1} a_rx^r $ then find the required value

Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$

$(1+x)^{20}=\sum_{r=1}^{20}a_rx^r$ where $a_r=\binom{20}{r}$, find the value $\mathop{\sum\sum}_{0\le i<j\le 20}(a_i-a_j)^2$

Binomial Theorem and Complex Numbers

Showing that percentage of the coefficients of $(x+y)^n$ being even tends towards $100\%$ when taking the limit of $n$ to infinity

Binomial theorem alternating sum

How do I find all the coefficients of $x^0$ in this: $(x-\frac{2}{\sqrt x})^8$

Compute $\binom n0 + 2\binom n1 + \binom n2 + 2\binom n3+...$

How do I prove $2 \binom n2 + 4 \binom n4 + 6 \binom n6 + \cdots = n 2^{n-2}$

Find the sum of the coefficients in the expansion of the given expression using Binomial Theorem.

Anti-symmetric ways

Sum of the coefficients of polynomial $f(x)= (3x-2)^{107} (x+1)^4$

Combinatorial proof (catching fishes and eating some from them)

General strategies for evaluating sums in probabilistic/combinatoric problems: $\sum_n\sum_m {n+m \choose n}p^{n+m}(1-p)^{n+m}$

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