Question

The sum $\sum_{k=0}^{n} (-1)^k \frac{{n \choose k}}{{n+k \choose k}}=\frac{1}{2}$

score 106 · Answer 1 · 2019-06-25 13:07:37

106

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The sum $\sum_{k=0}^{n} (-1)^k \frac{{n \choose k}}{{n+k \choose k}}=\frac{1}{2}$

Published on 25 Jun 2019 - 13:07

score 69 · Answer 2 · 2019-06-25 21:30:52

69

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Sum up to n terms of a series containing binomial coefficients

Published on 25 Jun 2019 - 21:30

score 57 · Answer 3 · 2019-06-27 07:52:59

57

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How to find $a$ and $n$ in the following equation $(1+ax)^n$ by using Pascal triangle?

Published on 27 Jun 2019 - 7:52

score 206 · Answer 4 · 2019-06-27 15:57:31

206

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Fraction with binomial coefficient

Published on 27 Jun 2019 - 15:57

score 560 · Answer 5 · 2019-06-29 04:07:33

A conjecture formula: $\sum\limits_{n=1}^\infty \frac{\binom{mn}{n}}{n}\left(\frac{(m-1)^{m-1}}{m^m} \right)^n=m\log\left(\frac{m}{m-1}\right)$

score 799 · Answer 6 · 2026-03-27 21:52:58

799

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A Binomial Identity simplify

Published on 27 Mar 2026 - 21:52

score 245 · Answer 7 · 2026-03-27 05:38:16

245

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On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$?

Published on 27 Mar 2026 - 5:38

score 121 · Answer 8 · 2019-07-03 16:21:03

121

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What are the coefficients of the polynomial expansion of binomial coefficients?

Published on 03 Jul 2019 - 16:21

score 75 · Answer 9 · 2019-07-04 17:02:11

75

Views

Sum of different combinations.

Published on 04 Jul 2019 - 17:02

score 87 · Answer 10 · 2019-07-05 19:25:07

87

Views

Why does $\binom{n}{k} = 0$, if $k > n$?

Published on 05 Jul 2019 - 19:25

score 157 · Answer 11 · 2019-07-05 23:14:20

157

Views

Prove that $\sum_{X=0}^N u(X) {N \choose X} p^X (1-p)^{N-X}=0 \iff u(X)=0, \space \forall X\in\{ 0,1,...,N \}$

Published on 05 Jul 2019 - 23:14

score 166 · Answer 12 · 2019-07-06 15:43:14

166

Views

How to create a formula to calculate the different combinations of money amount distribution?

Published on 06 Jul 2019 - 15:43

score 424 · Answer 13 · 2019-07-06 17:32:24

424

Views

A combinatorial identity involving square of central binomial coefficient.

Published on 06 Jul 2019 - 17:32

score 139 · Answer 14 · 2019-07-06 20:33:26

139

Views

Geometric interpretation of $\binom{n+2}{3}$ in Knuth's book

Published on 06 Jul 2019 - 20:33

score 109 · Answer 15 · 2019-07-07 11:14:57

109

Views

Prove the estimate $\binom{n}{k} \le (\frac{en}{k})^k$

Published on 07 Jul 2019 - 11:14

List Question

The sum $\sum_{k=0}^{n} (-1)^k \frac{{n \choose k}}{{n+k \choose k}}=\frac{1}{2}$

Sum up to n terms of a series containing binomial coefficients

How to find $a$ and $n$ in the following equation $(1+ax)^n$ by using Pascal triangle?

Fraction with binomial coefficient

A conjecture formula: $\sum\limits_{n=1}^\infty \frac{\binom{mn}{n}}{n}\left(\frac{(m-1)^{m-1}}{m^m} \right)^n=m\log\left(\frac{m}{m-1}\right)$

A Binomial Identity simplify

On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$?

What are the coefficients of the polynomial expansion of binomial coefficients?

Sum of different combinations.

Why does $\binom{n}{k} = 0$, if $k > n$?

Prove that $\sum_{X=0}^N u(X) {N \choose X} p^X (1-p)^{N-X}=0 \iff u(X)=0, \space \forall X\in\{ 0,1,...,N \}$

How to create a formula to calculate the different combinations of money amount distribution?

A combinatorial identity involving square of central binomial coefficient.

Geometric interpretation of $\binom{n+2}{3}$ in Knuth's book

Prove the estimate $\binom{n}{k} \le (\frac{en}{k})^k$

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