Question

Congruences involving central binomial coefficients

score 40 · Answer 1 · 2026-04-14 01:44:05

40

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Congruences involving central binomial coefficients

Published on 14 Apr 2026 - 1:44

score 36 · Answer 2 · 2026-04-15 00:51:08

36

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Sum : $\sum_{t=1}^{r} t\binom{n-t}{n-r}=\binom{n+1}{r-1}$

Published on 15 Apr 2026 - 0:51

score 68 · Answer 3 · 2026-04-16 17:13:18

68

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$\displaystyle\sum_{n=1}^{\infty}{\binom{v}{n}\dfrac{\Gamma\left({\large\frac{n}{2}}\right)}{\Gamma\left({\large\frac{n+1}{2}}\right)}}$

Published on 16 Apr 2026 - 17:13

score 137 · Answer 4 · 2026-04-09 11:29:15

137

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French Tarot suits probability

Published on 09 Apr 2026 - 11:29

score 112 · Answer 5 · 2026-04-14 16:18:43

112

Views

Summation of Alternating Products in Pascal's Triangle: prove $\sum_{k=1}^{n} (-1)^{k+1} x_{k} y_{n-k+1} = 1$

Published on 14 Apr 2026 - 16:18

score 94 · Answer 6 · 2026-04-14 15:38:58

94

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General formula for $\sum_{k=0}^n k^a \binom{n}{k}$ is somehow hypergeometric?

Published on 14 Apr 2026 - 15:38

score 106 · Answer 7 · 2026-04-13 01:57:39

106

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Identity involving binomial coefficients, Stirling numbers of the second kind, and generalized hypergeometric series

Published on 13 Apr 2026 - 1:57

score 45 · Answer 8 · 2026-04-10 19:54:18

45

Views

How to use binomial properties when proving by induction?

Published on 10 Apr 2026 - 19:54

score 153 · Answer 9 · 2026-04-16 00:48:41

153

Views

Proving (rigorously) that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$

Published on 16 Apr 2026 - 0:48

score 56 · Answer 10 · 2026-04-15 09:27:20

56

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Is $\lim_{m\to\infty}\sum_{k≥t}\frac{1}{\binom kt^m}=1$?

Published on 15 Apr 2026 - 9:27

score 88 · Answer 11 · 2026-04-11 16:55:03

88

Views

A combinational problem

Published on 11 Apr 2026 - 16:55

score 237 · Answer 12 · 2026-04-11 18:41:31

237

Views

Prove $\sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} {n \choose k} p^k (1-p)^{n-k}$

Published on 11 Apr 2026 - 18:41

score 90 · Answer 13 · 2026-04-13 15:57:10

90

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Binomial identity $\sum_{r=0}^n(2r+1)\binom{n}{r}^2=(n+1)\binom{2n}{n}$

Published on 13 Apr 2026 - 15:57

score 80 · Answer 14 · 2026-04-18 05:37:56

80

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Finding the coefficient of $x^0$ in $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1} - \frac{x-1}{x-x^{1/2}}\right)^{10}$

Published on 18 Apr 2026 - 5:37

score 56 · Answer 15 · 2026-04-12 06:25:56

56

Views

Correlation between binomial coefficients

Published on 12 Apr 2026 - 6:25

List Question

Congruences involving central binomial coefficients

Sum : $\sum_{t=1}^{r} t\binom{n-t}{n-r}=\binom{n+1}{r-1}$

$\displaystyle\sum_{n=1}^{\infty}{\binom{v}{n}\dfrac{\Gamma\left({\large\frac{n}{2}}\right)}{\Gamma\left({\large\frac{n+1}{2}}\right)}}$

French Tarot suits probability

Summation of Alternating Products in Pascal's Triangle: prove $\sum_{k=1}^{n} (-1)^{k+1} x_{k} y_{n-k+1} = 1$

General formula for $\sum_{k=0}^n k^a \binom{n}{k}$ is somehow hypergeometric?

Identity involving binomial coefficients, Stirling numbers of the second kind, and generalized hypergeometric series

How to use binomial properties when proving by induction?

Proving (rigorously) that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$

Is $\lim_{m\to\infty}\sum_{k≥t}\frac{1}{\binom kt^m}=1$?

A combinational problem

Prove $\sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} {n \choose k} p^k (1-p)^{n-k}$

Binomial identity $\sum_{r=0}^n(2r+1)\binom{n}{r}^2=(n+1)\binom{2n}{n}$

Finding the coefficient of $x^0$ in $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1} - \frac{x-1}{x-x^{1/2}}\right)^{10}$

Correlation between binomial coefficients

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