Question

Proof $\sum_{k=0}^\infty \binom{k}{n-k} = f_{n+1}$ where $f_n$ is the n-th Fibonacci-number

score 336 · Answer 1 · 2026-03-26 11:01:19

336

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Proof $\sum_{k=0}^\infty \binom{k}{n-k} = f_{n+1}$ where $f_n$ is the n-th Fibonacci-number

Published on 26 Mar 2026 - 11:01

score 2.8k · Answer 2 · 2019-01-25 22:55:36

2.8k

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How to prove binomial coefficient $ {2^n \choose k} $ is even number?

Published on 25 Jan 2019 - 22:55

score 152 · Answer 3 · 2019-01-27 04:54:37

152

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Prove that $\sum_{k=0}^n 2^k \binom{n}{k} \binom{n-k}{\lfloor (n-k)/2 \rfloor}=\binom{2n+1}{n}$

Published on 27 Jan 2019 - 4:54

score 299 · Answer 4 · 2019-01-27 16:49:25

299

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Why $\binom{mp-1}{p-1} \equiv 1 \pmod {p^3}$?

Published on 27 Jan 2019 - 16:49

score 169 · Answer 5 · 2026-03-27 10:34:19

169

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Summation of combination of binomial coefficient

Published on 27 Mar 2026 - 10:34

score 115 · Answer 6 · 2026-03-27 00:38:54

115

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Show that the cardinal of $ A := \left\{ k \in \mathbf{Z} | 0 \leq k \leq n \text{ and } \binom{n}{k} \text{ is odd} \right\} $ is a power of 2

Published on 27 Mar 2026 - 0:38

score 44 · Answer 7 · 2026-03-27 10:24:26

44

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Limit of a sum with binomial term, where terms do not add up to 1

Published on 27 Mar 2026 - 10:24

score 47 · Answer 8 · 2026-03-28 23:57:55

47

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Why is this sum simplified to this value?

Published on 28 Mar 2026 - 23:57

score 271 · Answer 9 · 2026-03-27 10:24:26

271

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Concerning the identity in sums of Binomial coefficients

Published on 27 Mar 2026 - 10:24

score 159 · Answer 10 · 2026-03-27 10:31:27

159

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On the parity of the coefficients of $(x+y)^n$.

Published on 27 Mar 2026 - 10:31

score 248 · Answer 11 · 2019-01-29 19:50:44

248

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More about odd numbers in Pascal's triangle

Published on 29 Jan 2019 - 19:50

score 330 · Answer 12 · 2019-02-01 07:08:12

330

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Is there a simple combinatoric interpretation of this identity?

Published on 01 Feb 2019 - 7:08

score 94 · Answer 13 · 2026-03-27 13:43:07

94

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Parity of some binomial coefficients

Published on 27 Mar 2026 - 13:43

score 273 · Answer 14 · 2019-02-02 10:47:25

273

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Alternating shifted central binomial sum with Cauchy weights

Published on 02 Feb 2019 - 10:47

score 130 · Answer 15 · 2026-03-25 22:23:46

130

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Is it valid to define $\binom{n}{n+k} = 0$

Published on 25 Mar 2026 - 22:23

List Question

Proof $\sum_{k=0}^\infty \binom{k}{n-k} = f_{n+1}$ where $f_n$ is the n-th Fibonacci-number

How to prove binomial coefficient $ {2^n \choose k} $ is even number?

Prove that $\sum_{k=0}^n 2^k \binom{n}{k} \binom{n-k}{\lfloor (n-k)/2 \rfloor}=\binom{2n+1}{n}$

Why $\binom{mp-1}{p-1} \equiv 1 \pmod {p^3}$?

Summation of combination of binomial coefficient

Show that the cardinal of $ A := \left\{ k \in \mathbf{Z} | 0 \leq k \leq n \text{ and } \binom{n}{k} \text{ is odd} \right\} $ is a power of 2

Limit of a sum with binomial term, where terms do not add up to 1

Why is this sum simplified to this value?

Concerning the identity in sums of Binomial coefficients

On the parity of the coefficients of $(x+y)^n$.

More about odd numbers in Pascal's triangle

Is there a simple combinatoric interpretation of this identity?

Parity of some binomial coefficients

Alternating shifted central binomial sum with Cauchy weights

Is it valid to define $\binom{n}{n+k} = 0$

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