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Analytically, why does Sierpinski's triangle appear in Pascal's triangle?

score 1.8k · Answer 1 · 2026-03-28 10:17:33

1.8k

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Analytically, why does Sierpinski's triangle appear in Pascal's triangle?

Published on 28 Mar 2026 - 10:17

score 166 · Answer 2 · 2026-04-05 07:18:32

166

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Closed formula for $\sum_{k=0}^{n}\binom{n+k}{k}x^k$

Published on 05 Apr 2026 - 7:18

score 47 · Answer 3 · 2020-03-05 17:37:49

47

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showing that $\binom{m^{3}}{m+1}= \binom{m^{3}}{m}\cdot(m^2-m)$

Published on 05 Mar 2020 - 17:37

score 33 · Answer 4 · 2020-03-05 20:35:12

33

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Factorial, binomial coefficient, arrange numbers

Published on 05 Mar 2020 - 20:35

score 173 · Answer 5 · 2020-03-06 10:05:36

173

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About the closed form of $\sum _{k=1}^{n}\binom{n}{k}\frac{1}{k}$

Published on 06 Mar 2020 - 10:05

score 202 · Answer 6 · 2020-03-06 18:44:28

202

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Prove $\sum_{k=0}^{n}\binom{2k}{k}\binom{2(n-k)}{n-k}\frac{1}{(2k+1)(2n-2k+1)}=\frac{2^{\left(4n+1\right)}}{\left(n+1\right)^{2}\binom{2n+2}{n+1}}$

Published on 06 Mar 2020 - 18:44

score 317 · Answer 7 · 2020-03-07 09:16:45

317

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How can you prove the Rothe–Hagen identity algebraically, not using Double Counting (combinatorics proof technique)?

Published on 07 Mar 2020 - 9:16

score 74 · Answer 8 · 2020-03-07 13:57:08

74

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Prove that $\binom{2}{2}+\binom{3}{2}+\binom{4}{2}+...+\binom{n}{2}=\binom{n+1}{3}$

Published on 07 Mar 2020 - 13:57

score 186 · Answer 9 · 2020-03-07 15:13:10

186

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3 binomial identities that I could not find a closed form for them

Published on 07 Mar 2020 - 15:13

score 133 · Answer 10 · 2026-04-01 15:57:04

133

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Closed-form expression of a series involving combinatorics

Published on 01 Apr 2026 - 15:57

score 264 · Answer 11 · 2026-04-01 15:49:39

264

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Somewhat challenging binomial identity

Published on 01 Apr 2026 - 15:49

score 94 · Answer 12 · 2020-03-08 15:26:02

94

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Closed form for $\sum_{k=0}^{n}\binom{\binom{n}{k}}{k}$

Published on 08 Mar 2020 - 15:26

score 139 · Answer 13 · 2020-03-09 10:55:53

139

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Closed form for two binomial identities

Published on 09 Mar 2020 - 10:55

score 112 · Answer 14 · 2020-03-09 16:05:54

112

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Problem with two binomial identities from Concrete mathematics book

Published on 09 Mar 2020 - 16:05

score 57 · Answer 15 · 2026-03-29 23:30:28

57

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All permutation of a set including the ones where some elements of the set are missing

Published on 29 Mar 2026 - 23:30

List Question

Analytically, why does Sierpinski's triangle appear in Pascal's triangle?

Closed formula for $\sum_{k=0}^{n}\binom{n+k}{k}x^k$

showing that $\binom{m^{3}}{m+1}= \binom{m^{3}}{m}\cdot(m^2-m)$

Factorial, binomial coefficient, arrange numbers

About the closed form of $\sum _{k=1}^{n}\binom{n}{k}\frac{1}{k}$

Prove $\sum_{k=0}^{n}\binom{2k}{k}\binom{2(n-k)}{n-k}\frac{1}{(2k+1)(2n-2k+1)}=\frac{2^{\left(4n+1\right)}}{\left(n+1\right)^{2}\binom{2n+2}{n+1}}$

How can you prove the Rothe–Hagen identity algebraically, not using Double Counting (combinatorics proof technique)?

Prove that $\binom{2}{2}+\binom{3}{2}+\binom{4}{2}+...+\binom{n}{2}=\binom{n+1}{3}$

3 binomial identities that I could not find a closed form for them

Closed-form expression of a series involving combinatorics

Somewhat challenging binomial identity

Closed form for $\sum_{k=0}^{n}\binom{\binom{n}{k}}{k}$

Closed form for two binomial identities

Problem with two binomial identities from Concrete mathematics book

All permutation of a set including the ones where some elements of the set are missing

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