Question

Binomial series sum of the form $\sum^{k}_{r=0}(-1)^r(2)^{k-r}\binom{20}{r}\binom{20-r}{20-k}$

score 165 · Answer 1 · 2026-04-13 19:10:31

165

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Binomial series sum of the form $\sum^{k}_{r=0}(-1)^r(2)^{k-r}\binom{20}{r}\binom{20-r}{20-k}$

Published on 13 Apr 2026 - 19:10

score 64 · Answer 2 · 2026-04-15 04:35:28

64

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$\sum_{i=1}^{k} \binom{n}{i} \le (\frac{en}{k})^k$

Published on 15 Apr 2026 - 4:35

score 477 · Answer 3 · 2026-04-13 13:51:55

477

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Signs of entries of Kravchuk matrices asymptotically produce a large circular region with hyperbolic sinks. Why?

Published on 13 Apr 2026 - 13:51

score 164 · Answer 4 · 2026-04-12 02:11:32

164

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Closed form for $\sum_{k=0}^{n} {n \choose k} \dfrac{2^{k+1}}{k+1}$

Published on 12 Apr 2026 - 2:11

score 83 · Answer 5 · 2026-03-25 23:09:07

83

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Probability / Permuations: Expected Number of Games Till Bust

Published on 25 Mar 2026 - 23:09

score 109 · Answer 6 · 2026-04-13 05:24:59

109

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Complex binomial series

Published on 13 Apr 2026 - 5:24

score 152 · Answer 7 · 2026-04-16 04:44:13

152

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Prove that $ \sum_{i=0}^n {n \choose i} \cdot (-1)^i \cdot i^n =(-1)^n n!$ and $ \sum_{i=0}^n {n \choose i} \cdot (-1)^i \cdot i^k = 0$ for $k<n$

Published on 16 Apr 2026 - 4:44

score 71 · Answer 8 · 2026-04-15 04:14:54

71

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Why does ${n \choose k }{k \choose l} = {n \choose l}{n - l \choose k - l}$

Published on 15 Apr 2026 - 4:14

score 195 · Answer 9 · 2026-04-12 11:38:26

195

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problem of binomial coefficients

Published on 12 Apr 2026 - 11:38

score 100 · Answer 10 · 2026-04-13 03:20:33

100

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Find the sum of $\sum_{k=1}^n k(k-1) {{n} \choose {k-1}}$

Published on 13 Apr 2026 - 3:20

score 169 · Answer 11 · 2026-04-16 18:55:30

169

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Proving that ${(1-\frac{2}{x^2})}^x < \frac{x-1}{x+1}$ for any $x > 2$.

Published on 16 Apr 2026 - 18:55

score 43 · Answer 12 · 2026-04-17 08:20:24

$\frac{\binom{i}{N}^2}{\binom{N}{2N}}\left(\frac{i}{N}\right)^2= \frac{\binom{i-1}{N}^2}{\binom{N}{2N}}\left(1-\frac{i-1}{N}\right)^2$

score 124 · Answer 13 · 2026-04-17 08:33:36

124

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Why $\bigg\lfloor \frac{n^m}{\binom{n}{m}}\bigg\rfloor=m!$ for $n>(m+1)^{m+2}$?

Published on 17 Apr 2026 - 8:33

score 99 · Answer 14 · 2026-03-25 06:11:12

99

Views

Identity involving q-Pochhammer symbols -- "Normalization"

Published on 25 Mar 2026 - 6:11

score 153 · Answer 15 · 2026-04-16 02:38:08

153

Views

Big-O of the log of sum of binomial coefficient

Published on 16 Apr 2026 - 2:38

List Question

Binomial series sum of the form $\sum^{k}_{r=0}(-1)^r(2)^{k-r}\binom{20}{r}\binom{20-r}{20-k}$

$\sum_{i=1}^{k} \binom{n}{i} \le (\frac{en}{k})^k$

Signs of entries of Kravchuk matrices asymptotically produce a large circular region with hyperbolic sinks. Why?

Closed form for $\sum_{k=0}^{n} {n \choose k} \dfrac{2^{k+1}}{k+1}$

Probability / Permuations: Expected Number of Games Till Bust

Complex binomial series

Prove that $ \sum_{i=0}^n {n \choose i} \cdot (-1)^i \cdot i^n =(-1)^n n!$ and $ \sum_{i=0}^n {n \choose i} \cdot (-1)^i \cdot i^k = 0$ for $k<n$

Why does ${n \choose k }{k \choose l} = {n \choose l}{n - l \choose k - l}$

problem of binomial coefficients

Find the sum of $\sum_{k=1}^n k(k-1) {{n} \choose {k-1}}$

Proving that ${(1-\frac{2}{x^2})}^x < \frac{x-1}{x+1}$ for any $x > 2$.

$\frac{\binom{i}{N}^2}{\binom{N}{2N}}\left(\frac{i}{N}\right)^2= \frac{\binom{i-1}{N}^2}{\binom{N}{2N}}\left(1-\frac{i-1}{N}\right)^2$

Why $\bigg\lfloor \frac{n^m}{\binom{n}{m}}\bigg\rfloor=m!$ for $n>(m+1)^{m+2}$?

Identity involving q-Pochhammer symbols -- "Normalization"

Big-O of the log of sum of binomial coefficient

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