Question

Show $\sum_{k=0}^{n}2^{n-k}\binom{a+k}{k}\frac{a-k}{a+k}=\binom{a+n}{n}$

score 114 · Answer 1 · 2020-03-13 11:28:04

114

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Show $\sum_{k=0}^{n}2^{n-k}\binom{a+k}{k}\frac{a-k}{a+k}=\binom{a+n}{n}$

Published on 13 Mar 2020 - 11:28

score 154 · Answer 2 · 2020-03-13 15:41:51

154

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Identity related to $\sum_{k=0}^{n}\frac{x^k}{\binom{n}{k}}$

Published on 13 Mar 2020 - 15:41

score 317 · Answer 3 · 2026-03-27 23:15:44

317

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Proof that $C(n , 0) + C(n+1 , 1) + C(n+2 , 2) + ... + C(n + k , k) =C(n+k+1 , k)$

Published on 27 Mar 2026 - 23:15

score 64 · Answer 4 · 2026-03-27 23:14:21

64

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How to prove that $\sum_{j=0}^k {n\choose j} \cdot{m\choose k -j} = {n+m\choose k}$

Published on 27 Mar 2026 - 23:14

score 323 · Answer 5 · 2026-03-30 04:24:53

323

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Evaluate $\sum_{k=0}^{50}{{100}\choose{2k}}18^{2k}$

Published on 30 Mar 2026 - 4:24

score 88 · Answer 6 · 2026-03-25 23:42:17

88

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What is the sum this series involving sum of reciprocals of natural numbers?

Published on 25 Mar 2026 - 23:42

score 38 · Answer 7 · 2026-03-26 19:37:52

38

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Finite Geometric Series Scaled by Binomial Coefficients

Published on 26 Mar 2026 - 19:37

score 68 · Answer 8 · 2020-03-15 16:36:29

68

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How can it be shown that $\sum_{k=0}^{n}\binom{n}{k}\binom{2n-k}{n}\left(a-1\right)^k=\sum_{k=0}^{n}\binom{n}{k}^{2}a^{n-k}$

Published on 15 Mar 2020 - 16:36

score 49 · Answer 9 · 2020-03-15 17:22:05

49

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Series with reciprocal of binomial coefficients

Published on 15 Mar 2020 - 17:22

score 94 · Answer 10 · 2020-03-15 18:25:28

94

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Prove alternating sum of binomial coefficients squared by induction

Published on 15 Mar 2020 - 18:25

score 102 · Answer 11 · 2026-03-30 22:56:12

102

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Closed form for $\sum_{k=0}^{n}\binom{n}{k}^{r}$

Published on 30 Mar 2026 - 22:56

score 245 · Answer 12 · 2020-03-16 03:06:53

245

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Evaluate the sum $\binom{2n}{n}+\binom{3n}{n}+\binom{4n}{n}+\cdots+\binom{kn}{n}$

Published on 16 Mar 2020 - 3:06

score 104 · Answer 13 · 2020-03-16 09:12:46

104

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Prove $\sum_{k=0}^{r}\binom{n}{k}\binom{n}{r-k}\left(-1\right)^{k}=\left(-1\right)^{\frac{r}{2}}\binom{n}{\frac{r}{2}}$ for $r$ even.

Published on 16 Mar 2020 - 9:12

score 296 · Answer 14 · 2026-03-30 04:25:14

296

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Solving binomial summation $\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$

Published on 30 Mar 2026 - 4:25

score 177 · Answer 15 · 2020-03-16 17:13:58

177

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Looking for a short proof of a harmless looking binomial identity

Published on 16 Mar 2020 - 17:13

List Question

Show $\sum_{k=0}^{n}2^{n-k}\binom{a+k}{k}\frac{a-k}{a+k}=\binom{a+n}{n}$

Identity related to $\sum_{k=0}^{n}\frac{x^k}{\binom{n}{k}}$

Proof that $C(n , 0) + C(n+1 , 1) + C(n+2 , 2) + ... + C(n + k , k) =C(n+k+1 , k)$

How to prove that $\sum_{j=0}^k {n\choose j} \cdot{m\choose k -j} = {n+m\choose k}$

Evaluate $\sum_{k=0}^{50}{{100}\choose{2k}}18^{2k}$

What is the sum this series involving sum of reciprocals of natural numbers?

Finite Geometric Series Scaled by Binomial Coefficients

How can it be shown that $\sum_{k=0}^{n}\binom{n}{k}\binom{2n-k}{n}\left(a-1\right)^k=\sum_{k=0}^{n}\binom{n}{k}^{2}a^{n-k}$

Series with reciprocal of binomial coefficients

Prove alternating sum of binomial coefficients squared by induction

Closed form for $\sum_{k=0}^{n}\binom{n}{k}^{r}$

Evaluate the sum $\binom{2n}{n}+\binom{3n}{n}+\binom{4n}{n}+\cdots+\binom{kn}{n}$

Prove $\sum_{k=0}^{r}\binom{n}{k}\binom{n}{r-k}\left(-1\right)^{k}=\left(-1\right)^{\frac{r}{2}}\binom{n}{\frac{r}{2}}$ for $r$ even.

Solving binomial summation $\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$

Looking for a short proof of a harmless looking binomial identity

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