Question

combinatorial proof for an identity with generating functions $\frac{1}{(1-x)^n} = \sum_{r=0}^\infty{r+n-1 \choose r}x^r$

score 208 · Answer 1 · 2026-03-26 06:34:58

208

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combinatorial proof for an identity with generating functions $\frac{1}{(1-x)^n} = \sum_{r=0}^\infty{r+n-1 \choose r}x^r$

Published on 26 Mar 2026 - 6:34

score 33 · Answer 2 · 2020-04-11 20:11:03

33

Views

Binomial coefficient, question

Published on 11 Apr 2020 - 20:11

score 305 · Answer 3 · 2026-04-03 16:22:23

305

Views

Is there a trick for adding and subtracting binomial coefficients?

Published on 03 Apr 2026 - 16:22

score 85 · Answer 4 · 2026-03-25 09:27:28

85

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Geometric series with product of binomial coefficents

Published on 25 Mar 2026 - 9:27

score 36 · Answer 5 · 2020-04-12 08:25:14

36

Views

On powers of binomial coefficient

Published on 12 Apr 2020 - 8:25

score 192 · Answer 6 · 2020-04-12 09:39:49

192

Views

How to prove $\sum^n_{k=0}\binom{n}k\cos\big((n-2k)\theta\big)=2^n\cos^n\theta$?

Published on 12 Apr 2020 - 9:39

score 119 · Answer 7 · 2020-04-12 23:47:37

119

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Binomial coefficient - upper factorials

Published on 12 Apr 2020 - 23:47

score 244 · Answer 8 · 2026-03-27 03:44:20

244

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Identity of binomial coefficients: $\binom{n+1}{k+1}=\sum_{i=m}^{n+m-k}{\binom{n-i}{k-m}\binom{i}{m}}$

Published on 27 Mar 2026 - 3:44

score 83 · Answer 9 · 2026-03-30 04:38:33

83

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Combinatorial identity: Lah numbers as sum with binomials

Published on 30 Mar 2026 - 4:38

score 276 · Answer 10 · 2020-04-14 08:22:10

276

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Summing the even numbers in any row of pascal's triangle.

Published on 14 Apr 2020 - 8:22

score 186 · Answer 11 · 2020-04-14 12:52:23

186

Views

The identity $\binom{n+k}{k} = \sum_i \binom{n}{i}\binom{k}{i}$

Published on 14 Apr 2020 - 12:52

score 52 · Answer 12 · 2026-03-30 07:38:58

52

Views

A proof of Bernouilli's inequality

Published on 30 Mar 2026 - 7:38

score 495 · Answer 13 · 2020-04-14 18:30:37

495

Views

Triple Summation of product of Binomial Coefficients $\sum_{k=0}^n\sum_{j=0}^n\sum_{i=0}^n\binom ni\binom{n-i}j\binom{n-j}k=5^n$?

Published on 14 Apr 2020 - 18:30

score 193 · Answer 14 · 2026-03-28 09:33:42

193

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Closed form of $\sum_{k=0}^{\infty} \sum_{m=0}^{\infty} r^m \cdot t^k \binom{m+k}{k} \binom{m+k+1}{k}$ for fixed $r, t$

Published on 28 Mar 2026 - 9:33

score 101 · Answer 15 · 2020-04-16 21:04:02

101

Views

Probability Theory : The Logic of Science Equation 6.16

Published on 16 Apr 2020 - 21:04

List Question

combinatorial proof for an identity with generating functions $\frac{1}{(1-x)^n} = \sum_{r=0}^\infty{r+n-1 \choose r}x^r$

Binomial coefficient, question

Is there a trick for adding and subtracting binomial coefficients?

Geometric series with product of binomial coefficents

On powers of binomial coefficient

How to prove $\sum^n_{k=0}\binom{n}k\cos\big((n-2k)\theta\big)=2^n\cos^n\theta$?

Binomial coefficient - upper factorials

Identity of binomial coefficients: $\binom{n+1}{k+1}=\sum_{i=m}^{n+m-k}{\binom{n-i}{k-m}\binom{i}{m}}$

Combinatorial identity: Lah numbers as sum with binomials

Summing the even numbers in any row of pascal's triangle.

The identity $\binom{n+k}{k} = \sum_i \binom{n}{i}\binom{k}{i}$

A proof of Bernouilli's inequality

Triple Summation of product of Binomial Coefficients $\sum_{k=0}^n\sum_{j=0}^n\sum_{i=0}^n\binom ni\binom{n-i}j\binom{n-j}k=5^n$?

Closed form of $\sum_{k=0}^{\infty} \sum_{m=0}^{\infty} r^m \cdot t^k \binom{m+k}{k} \binom{m+k+1}{k}$ for fixed $r, t$

Probability Theory : The Logic of Science Equation 6.16

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