Question

Closed form solution for $\sum_{j=0}^{n} {n \choose j}^{2}x^{j}$?

score 68 · Answer 1 · 2019-05-28 01:03:01

68

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Closed form solution for $\sum_{j=0}^{n} {n \choose j}^{2}x^{j}$?

Published on 28 May 2019 - 1:03

score 87 · Answer 2 · 2026-03-27 21:43:27

87

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Harmonic series times binomial

Published on 27 Mar 2026 - 21:43

score 74 · Answer 3 · 2026-03-26 06:10:55

74

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Is there an infinity of $k \in \mathbb{N}$ s.t. $2 {k \choose k/2} - 1$ is prime?

Published on 26 Mar 2026 - 6:10

score 149 · Answer 4 · 2026-03-26 01:29:10

149

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Proof of $\sum_{k=1}^{i} \binom{n-k}{n-i} = \frac{n \cdot \binom{n-1}{n-i}}{n+1-i}$

Published on 26 Mar 2026 - 1:29

score 60 · Answer 5 · 2019-05-30 13:21:21

60

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Combinatorial identity with numerator doubled in binomial coefficient

Published on 30 May 2019 - 13:21

score 78 · Answer 6 · 2019-05-30 15:58:25

78

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Prove that $\sum_{0\leq k\leq r} \binom{n+k}{k} \binom{m+n+k}{n+k} = \binom{m+n}{n} \binom{m+n+r+1}{m+n+1}$ where $m,n,r\in \mathbb{N} $

Published on 30 May 2019 - 15:58

score 89 · Answer 7 · 2019-05-31 10:37:45

89

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Prove that $\sum_{k=0}^{n} \frac{{m \choose k}}{{n \choose k}}= \frac{n+1}{n-m+1}, m \le n$

Published on 31 May 2019 - 10:37

score 107 · Answer 8 · 2026-03-27 21:55:48

107

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How can I utilize the binomial expansion hint for part ii)?

Published on 27 Mar 2026 - 21:55

score 565 · Answer 9 · 2026-03-26 22:15:23

565

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How many tuples of {$a, b, c, \ldots$} satisfy $a+b+c+\ldots \leq n$?

Published on 26 Mar 2026 - 22:15

score 126 · Answer 10 · 2026-03-27 21:52:43

126

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binomial expansion questions

Published on 27 Mar 2026 - 21:52

score 218 · Answer 11 · 2019-06-01 17:56:18

218

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Proving that ${n+3\choose 3} =\frac{n+2}{2}\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$

Published on 01 Jun 2019 - 17:56

score 293 · Answer 12 · 2026-03-27 03:42:14

293

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Closed form for $\sum_{n=1}^\infty \frac{4^n}{n^p\binom{2n}{n}}$

Published on 27 Mar 2026 - 3:42

score 327 · Answer 13 · 2019-06-03 16:48:11

327

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Alternating sum of binomial coefficients multiplied by index to an n-2 extent

Published on 03 Jun 2019 - 16:48

score 148 · Answer 14 · 2019-06-04 07:01:19

148

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Summing $S_{n,m}=\sum_{k=1}^{n} (-1)^k~ k^{m} ~ {n \choose k}$ for $m<n,=,>n.$

Published on 04 Jun 2019 - 7:01

score 181 · Answer 15 · 2019-06-04 08:52:13

181

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A determinant of a matrix with binomial coefficients

Published on 04 Jun 2019 - 8:52

List Question

Closed form solution for $\sum_{j=0}^{n} {n \choose j}^{2}x^{j}$?

Harmonic series times binomial

Is there an infinity of $k \in \mathbb{N}$ s.t. $2 {k \choose k/2} - 1$ is prime?

Proof of $\sum_{k=1}^{i} \binom{n-k}{n-i} = \frac{n \cdot \binom{n-1}{n-i}}{n+1-i}$

Combinatorial identity with numerator doubled in binomial coefficient

Prove that $\sum_{0\leq k\leq r} \binom{n+k}{k} \binom{m+n+k}{n+k} = \binom{m+n}{n} \binom{m+n+r+1}{m+n+1}$ where $m,n,r\in \mathbb{N} $

Prove that $\sum_{k=0}^{n} \frac{{m \choose k}}{{n \choose k}}= \frac{n+1}{n-m+1}, m \le n$

How can I utilize the binomial expansion hint for part ii)?

How many tuples of {$a, b, c, \ldots$} satisfy $a+b+c+\ldots \leq n$?

binomial expansion questions

Proving that ${n+3\choose 3} =\frac{n+2}{2}\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$

Closed form for $\sum_{n=1}^\infty \frac{4^n}{n^p\binom{2n}{n}}$

Alternating sum of binomial coefficients multiplied by index to an n-2 extent

Summing $S_{n,m}=\sum_{k=1}^{n} (-1)^k~ k^{m} ~ {n \choose k}$ for $m<n,=,>n.$

A determinant of a matrix with binomial coefficients

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