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A peculiar binomial coefficient identity

score 355 · Answer 1 · 2013-06-26 21:18:46

355

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A peculiar binomial coefficient identity

Published on 26 Jun 2013 - 21:18

score 260 · Answer 2 · 2013-07-04 12:17:55

260

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Seeking a combinatorial proof of the identity$n+(n-1)2^1+(n-2)2^2+\cdots+ 2\cdot 2^{n-2}+ 1\cdot 2^{n-1}=-n+2^{n+1}-2$

Published on 04 Jul 2013 - 12:17

score 1.8k · Answer 3 · 2026-03-28 15:46:09

1.8k

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Stirling numbers combinatorial proof: $S(m,n)=\frac 1{n!} \sum_{k=0}^{n} (-1)^k\binom nk (n-k)^m$

Published on 28 Mar 2026 - 15:46

score 253 · Answer 4 · 2013-07-05 12:43:38

253

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Seeking a combinatorial proof of the identity$1 f_1+2 f_2+\cdots+n f_n=n f_{n + 2} - f_{n + 3} + 2$

Published on 05 Jul 2013 - 12:43

score 190 · Answer 5 · 2013-07-19 01:32:47

190

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Seeking a combinatorial proof of the identity $1^3+2^3+3^3+\cdots+n^3=\{{}_{n+1}C_{2}\}^2$

Published on 19 Jul 2013 - 1:32

score 1.1k · Answer 6 · 2013-08-03 05:11:02

1.1k

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Probability that number of heads flipped is divisible by 3

Published on 03 Aug 2013 - 5:11

score 174 · Answer 7 · 2013-08-14 13:16:30

174

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proofs for combinatorial identities?

Published on 14 Aug 2013 - 13:16

score 5.2k · Answer 8 · 2013-09-18 11:12:42

5.2k

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Fermat's Combinatorial Identity: How to prove combinatorially?

Published on 18 Sep 2013 - 11:12

score 2.6k · Answer 9 · 2026-03-28 17:25:55

2.6k

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Combinatorial Proof for Series of Stirling Numbers & Binomial Coefficients

Published on 28 Mar 2026 - 17:25

score 1k · Answer 10 · 2026-03-26 00:59:54

1k

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Combinatorial interpretation of an alternating binomial sum

Published on 26 Mar 2026 - 0:59

score 716 · Answer 11 · 2013-10-06 02:00:58

716

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Seeking a combinatorial proof of the identity$1+3+\cdots+(2n-1)=n^2$

Published on 06 Oct 2013 - 2:00

score 40.9k · Answer 12 · 2013-10-16 19:57:49

40.9k

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Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$

Published on 16 Oct 2013 - 19:57

score 720 · Answer 13 · 2013-11-08 00:41:38

720

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Strange combinatorial identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\binom{2n-2k}{n-1}=0$

Published on 08 Nov 2013 - 0:41

score 3k · Answer 14 · 2013-11-15 07:27:48

3k

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For all positive integers n,m,k where $n\ge m\ge k$ , $\binom {n}{m}\binom {m}{k}=\binom {n}{k}\binom {n-k}{n-m}$

Published on 15 Nov 2013 - 7:27

score 10.7k · Answer 15 · 2013-11-16 07:10:40

10.7k

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Combinatorial Proof of Multinomial Theorem - Without Induction or Binomial Theorem

Published on 16 Nov 2013 - 7:10

List Question

A peculiar binomial coefficient identity

Seeking a combinatorial proof of the identity$n+(n-1)2^1+(n-2)2^2+\cdots+ 2\cdot 2^{n-2}+ 1\cdot 2^{n-1}=-n+2^{n+1}-2$

Stirling numbers combinatorial proof: $S(m,n)=\frac 1{n!} \sum_{k=0}^{n} (-1)^k\binom nk (n-k)^m$

Seeking a combinatorial proof of the identity$1 f_1+2 f_2+\cdots+n f_n=n f_{n + 2} - f_{n + 3} + 2$

Seeking a combinatorial proof of the identity $1^3+2^3+3^3+\cdots+n^3=\{{}_{n+1}C_{2}\}^2$

Probability that number of heads flipped is divisible by 3

proofs for combinatorial identities?

Fermat's Combinatorial Identity: How to prove combinatorially?

Combinatorial Proof for Series of Stirling Numbers & Binomial Coefficients

Combinatorial interpretation of an alternating binomial sum

Seeking a combinatorial proof of the identity$1+3+\cdots+(2n-1)=n^2$

Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$

Strange combinatorial identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\binom{2n-2k}{n-1}=0$

For all positive integers n,m,k where $n\ge m\ge k$ , $\binom {n}{m}\binom {m}{k}=\binom {n}{k}\binom {n-k}{n-m}$

Combinatorial Proof of Multinomial Theorem - Without Induction or Binomial Theorem

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