Question

Asymptotic formula/closed form of $\sum_{r=0}^{n}\frac{(-1)^r\binom{n}{r}\log(n+r+1) }{n+r+1}$

score 122 · Answer 1 · 2026-04-14 10:48:11

122

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Asymptotic formula/closed form of $\sum_{r=0}^{n}\frac{(-1)^r\binom{n}{r}\log(n+r+1) }{n+r+1}$

Published on 14 Apr 2026 - 10:48

score 80 · Answer 2 · 2026-04-14 10:51:15

80

Views

Find an integer solution satisfying the following equality

Published on 14 Apr 2026 - 10:51

score 133 · Answer 3 · 2026-04-13 18:56:50

133

Views

Star of David theorem in Pascal's pyramid

Published on 13 Apr 2026 - 18:56

score 129 · Answer 4 · 2026-04-15 22:23:26

129

Views

Strengthening the Log-Concavity of Binomial Coefficients: $\binom{n}{k-1}\binom{n}{k+1} < \binom{n}{k}^2 - \binom{n}{k}$

Published on 15 Apr 2026 - 22:23

score 242 · Answer 5 · 2026-04-15 17:17:42

242

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Is $p\geq 3$ a prime divisor of $\sum_{t=0}^{\frac{p-1}{m}-1}\sum_{s=0}^{\frac{p-1}{m}-1}\binom{tm}{sm}$ where $m$ divides $p-1~?$

Published on 15 Apr 2026 - 17:17

score 251 · Answer 6 · 2026-04-16 03:21:51

251

Views

Partial fraction decomposition of $\frac{1}{(x(x+1)(x+2)...(x+n))^2}$

Published on 16 Apr 2026 - 3:21

score 31 · Answer 7 · 2026-04-16 03:32:55

31

Views

Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers

Published on 16 Apr 2026 - 3:32

score 91 · Answer 8 · 2026-04-13 20:16:27

91

Views

Closed form for $\sum_{k=0}^{j}\binom{n}{k}^2 x^k$

Published on 13 Apr 2026 - 20:16

score 52 · Answer 9 · 2026-04-14 22:30:41

52

Views

Binomial congruence mod $p^3$

Published on 14 Apr 2026 - 22:30

score 115 · Answer 10 · 2026-04-16 04:30:36

115

Views

$\sum_{k=0}^n{\binom{n}{k}\binom{n+k}{k}} = 0 \pmod 3$ iff there is at least a $1$ in the ternary expansion of $n$

Published on 16 Apr 2026 - 4:30

score 53 · Answer 11 · 2026-04-16 12:28:17

53

Views

Question about the binomial coefficent why is ${1} \choose {i-1}$ $=0$ if and only if $i-1>1$?

Published on 16 Apr 2026 - 12:28

score 306 · Answer 12 · 2026-04-12 18:49:09

306

Views

Prove of Vandermonde's identity

Published on 12 Apr 2026 - 18:49

score 39 · Answer 13 · 2026-04-14 13:41:48

39

Views

Is binomial distribution measuring combinations or successes?

Published on 14 Apr 2026 - 13:41

score 119 · Answer 14 · 2026-04-14 10:40:18

119

Views

About the value of $\sum_{k=0}^n \binom{n}{k}(-1)^k\frac{(n-k)^n}{n!}$

Published on 14 Apr 2026 - 10:40

score 60 · Answer 15 · 2026-04-16 18:47:12

60

Views

Proof that if $p^r$ divides $\binom{2n}{n}$ then $p^r\le2n$.

Published on 16 Apr 2026 - 18:47

List Question

Asymptotic formula/closed form of $\sum_{r=0}^{n}\frac{(-1)^r\binom{n}{r}\log(n+r+1) }{n+r+1}$

Find an integer solution satisfying the following equality

Star of David theorem in Pascal's pyramid

Strengthening the Log-Concavity of Binomial Coefficients: $\binom{n}{k-1}\binom{n}{k+1} < \binom{n}{k}^2 - \binom{n}{k}$

Is $p\geq 3$ a prime divisor of $\sum_{t=0}^{\frac{p-1}{m}-1}\sum_{s=0}^{\frac{p-1}{m}-1}\binom{tm}{sm}$ where $m$ divides $p-1~?$

Partial fraction decomposition of $\frac{1}{(x(x+1)(x+2)...(x+n))^2}$

Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers

Closed form for $\sum_{k=0}^{j}\binom{n}{k}^2 x^k$

Binomial congruence mod $p^3$

$\sum_{k=0}^n{\binom{n}{k}\binom{n+k}{k}} = 0 \pmod 3$ iff there is at least a $1$ in the ternary expansion of $n$

Question about the binomial coefficent why is ${1} \choose {i-1}$ $=0$ if and only if $i-1>1$?

Prove of Vandermonde's identity

Is binomial distribution measuring combinations or successes?

About the value of $\sum_{k=0}^n \binom{n}{k}(-1)^k\frac{(n-k)^n}{n!}$

Proof that if $p^r$ divides $\binom{2n}{n}$ then $p^r\le2n$.

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