Question

Fibonacci sum for $\pi$: $\sum_{n=1}^\infty\frac{F_{2n}}{n^2\binom{2n}{n}}=\frac{4\pi^2}{25\sqrt5}$

score 172 · Answer 1 · 2026-03-27 11:49:00

172

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Fibonacci sum for $\pi$: $\sum_{n=1}^\infty\frac{F_{2n}}{n^2\binom{2n}{n}}=\frac{4\pi^2}{25\sqrt5}$

Published on 27 Mar 2026 - 11:49

score 263 · Answer 2 · 2019-12-11 11:36:05

263

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Intuitive explanation of "reverse" binomial coefficient formula ${n\choose k}=\sum_{r=0}^s {n-s \choose k-r}{s\choose r}$

Published on 11 Dec 2019 - 11:36

score 224 · Answer 3 · 2026-03-29 16:21:26

224

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Proving that $\int_{0}^{1}x^{n-1}(1-x)^n \mathrm dx =\frac{1}{n {2n\choose n}}$

Published on 29 Mar 2026 - 16:21

score 143 · Answer 4 · 2019-12-13 15:16:01

143

Views

binomial identity: elementary proof possible?

Published on 13 Dec 2019 - 15:16

score 103 · Answer 5 · 2026-03-30 01:47:06

103

Views

Infinite sum involving $\frac{\binom{2n}{n}}{2^{2n}}$ resulting in gamma function product

Published on 30 Mar 2026 - 1:47

score 47 · Answer 6 · 2019-12-17 09:41:19

47

Views

Amount of Compositions of a Number n into k parts when the components are limited to the range [1;m] with m<n

Published on 17 Dec 2019 - 9:41

score 558 · Answer 7 · 2026-03-27 02:34:38

558

Views

Proof for any positive integer n the following equality hold

Published on 27 Mar 2026 - 2:34

score 54 · Answer 8 · 2019-12-19 10:25:51

54

Views

Derivation of binomial coefficient from second order difference equation

Published on 19 Dec 2019 - 10:25

score 152 · Answer 9 · 2026-03-29 20:21:26

152

Views

Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$

Published on 29 Mar 2026 - 20:21

score 314 · Answer 10 · 2019-12-20 18:36:30

314

Views

Compute telescopic sum of binomial coefficients

Published on 20 Dec 2019 - 18:36

score 327 · Answer 11 · 2026-03-27 04:22:00

327

Views

Is there a closed form for $\sum_{n=1}^\infty\frac{(-4)^nH_{n-1}^3}{{2n\choose n}n^2}\ ?$

Published on 27 Mar 2026 - 4:22

score 1.5k · Answer 12 · 2019-12-22 14:10:21

1.5k

Views

Binomial coefficients with real number

Published on 22 Dec 2019 - 14:10

score 117 · Answer 13 · 2026-03-29 18:02:07

117

Views

Proving an identity involving binomial coefficients

Published on 29 Mar 2026 - 18:02

score 36 · Answer 14 · 2026-03-29 17:53:13

36

Views

Convex combination with binomial probabilities

Published on 29 Mar 2026 - 17:53

score 113 · Answer 15 · 2019-12-24 09:23:24

113

Views

Finding a formula for a triangle similar to Pascal's.

Published on 24 Dec 2019 - 9:23

List Question

Fibonacci sum for $\pi$: $\sum_{n=1}^\infty\frac{F_{2n}}{n^2\binom{2n}{n}}=\frac{4\pi^2}{25\sqrt5}$

Intuitive explanation of "reverse" binomial coefficient formula ${n\choose k}=\sum_{r=0}^s {n-s \choose k-r}{s\choose r}$

Proving that $\int_{0}^{1}x^{n-1}(1-x)^n \mathrm dx =\frac{1}{n {2n\choose n}}$

binomial identity: elementary proof possible?

Infinite sum involving $\frac{\binom{2n}{n}}{2^{2n}}$ resulting in gamma function product

Amount of Compositions of a Number n into k parts when the components are limited to the range [1;m] with m<n

Proof for any positive integer n the following equality hold

Derivation of binomial coefficient from second order difference equation

Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$

Compute telescopic sum of binomial coefficients

Is there a closed form for $\sum_{n=1}^\infty\frac{(-4)^nH_{n-1}^3}{{2n\choose n}n^2}\ ?$

Binomial coefficients with real number

Proving an identity involving binomial coefficients

Convex combination with binomial probabilities

Finding a formula for a triangle similar to Pascal's.

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