Question

Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$

score 432 · Answer 1 · 2026-03-28 08:50:12

432

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Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$

Published on 28 Mar 2026 - 8:50

score 1.5k · Answer 2 · 2018-12-09 07:53:00

1.5k

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Proof for sum of digits of a number until sum is a single number

Published on 09 Dec 2018 - 7:53

score 156 · Answer 3 · 2026-03-30 01:52:08

156

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The series $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+...$

Published on 30 Mar 2026 - 1:52

score 43 · Answer 4 · 2018-12-09 17:02:39

43

Views

Fredholm integral equation needs to be written as a sum of functions

Published on 09 Dec 2018 - 17:02

score 81 · Answer 5 · 2018-12-09 17:58:16

81

Views

Closed form for $\sum_{n=1}^\infty \log(n) * x^n$

Published on 09 Dec 2018 - 17:58

score 112 · Answer 6 · 2026-03-28 12:14:53

112

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How to compute $\sum_{j=1}^{n} {{n}\choose {j}} \cdot (-1)^{j+1} \cdot \frac{(n-j)^{n-1}}{n^{n-1}}$

Published on 28 Mar 2026 - 12:14

score 40 · Answer 7 · 2026-03-30 20:52:57

40

Views

Finite sum of $1/n^{\delta}$

Published on 30 Mar 2026 - 20:52

score 140 · Answer 8 · 2026-03-26 03:13:32

140

Views

Show that $\limsup_{k \to \infty} 2^{-k} N_k = 0$ where $N_k$ is the number of $a_n \geq 2^{-k}$.

Published on 26 Mar 2026 - 3:13

score 26 · Answer 9 · 2026-03-25 17:38:20

26

Views

Trouble with changing indexing in summation

Published on 25 Mar 2026 - 17:38

score 150 · Answer 10 · 2026-03-27 20:19:57

150

Views

Evaluate $\sum_{n=1}^N n {n \choose k}$ and get a closed form solution

Published on 27 Mar 2026 - 20:19

score 31 · Answer 11 · 2026-04-01 04:20:25

31

Views

What does the notation of $1\{a_i=b\}$ mean in a summation?

Published on 01 Apr 2026 - 4:20

score 343 · Answer 12 · 2026-03-28 10:36:00

343

Views

Proof for this binomial coefficient's equation

Published on 28 Mar 2026 - 10:36

score 74 · Answer 13 · 2018-12-12 13:56:27

74

Views

Sum of All Combinations of Products of Two Matrices

Published on 12 Dec 2018 - 13:56

score 33 · Answer 14 · 2018-12-12 15:19:36

33

Views

$\sum_{i=0}^{2n-1} e^\frac{i}{n}$?

Published on 12 Dec 2018 - 15:19

score 210 · Answer 15 · 2018-12-12 16:05:51

210

Views

Sum to the nth term of an arithmetic-geometric series

Published on 12 Dec 2018 - 16:05

List Question

Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$

Proof for sum of digits of a number until sum is a single number

The series $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+...$

Fredholm integral equation needs to be written as a sum of functions

Closed form for $\sum_{n=1}^\infty \log(n) * x^n$

How to compute $\sum_{j=1}^{n} {{n}\choose {j}} \cdot (-1)^{j+1} \cdot \frac{(n-j)^{n-1}}{n^{n-1}}$

Finite sum of $1/n^{\delta}$

Show that $\limsup_{k \to \infty} 2^{-k} N_k = 0$ where $N_k$ is the number of $a_n \geq 2^{-k}$.

Trouble with changing indexing in summation

Evaluate $\sum_{n=1}^N n {n \choose k}$ and get a closed form solution

What does the notation of $1\{a_i=b\}$ mean in a summation?

Proof for this binomial coefficient's equation

Sum of All Combinations of Products of Two Matrices

$\sum_{i=0}^{2n-1} e^\frac{i}{n}$?

Sum to the nth term of an arithmetic-geometric series

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