Question

Bijective proof for $\sum\limits_{\substack{n_1+n_2=n\\m_1+m_2=m}}\binom{n_1+m_1}{n_1}^2\binom{n_2+m_2}{n_2}^2=\frac{1}{2}\binom{2n+2m+2}{2n+1}$?

score 94 · Answer 1 · 2026-04-16 13:09:47

94

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Bijective proof for $\sum\limits_{\substack{n_1+n_2=n\\m_1+m_2=m}}\binom{n_1+m_1}{n_1}^2\binom{n_2+m_2}{n_2}^2=\frac{1}{2}\binom{2n+2m+2}{2n+1}$?

Published on 16 Apr 2026 - 13:09

score 84 · Answer 2 · 2026-04-13 09:15:40

84

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Simplifying a double summation related to a Ramanujan-type pi formula $1/\pi$?

Published on 13 Apr 2026 - 9:15

score 111 · Answer 3 · 2026-04-08 07:48:09

111

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How to obtain explicit formula of coefficients for generating functions?

Published on 08 Apr 2026 - 7:48

score 65 · Answer 4 · 2026-04-16 06:49:09

65

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How to prove $\sum_{n\ge 0} a_n x^n = \sum_{n\ge 0} \left(\sum_{k=0}^{n}(-1)^{k+1} \binom{n}{k}a_k\right) \frac{x^n}{(x-1)^{n+1}}$?

Published on 16 Apr 2026 - 6:49

score 58 · Answer 5 · 2026-04-12 21:55:12

58

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score 84 · Answer 6 · 2026-04-15 11:27:17

84

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score 348 · Answer 7 · 2026-03-26 06:03:05

348

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score 93 · Answer 8 · 2026-04-14 15:24:00

93

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score 68 · Answer 9 · 2026-04-14 11:13:31

68

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Closed formula for $\sum_{k=0}^{\lfloor n/2\rfloor} x^k\binom{n-k}{k}$

Published on 14 Apr 2026 - 11:13

score 103 · Answer 10 · 2026-04-16 15:18:01

103

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Proving the identity $\sum_{k=0}^{m} \binom{n-k}{m-k}= \binom{n+1}{m}$ .

Published on 16 Apr 2026 - 15:18

score 55 · Answer 11 · 2026-04-12 09:59:18

55

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Alternating sum of products of half-shifted binomial coefficients

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score 142 · Answer 12 · 2026-04-10 01:30:54

142

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score 173 · Answer 13 · 2026-04-17 02:49:29

173

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Proving the identity $ \sum_{k=0}^{\min\{p,q\}} \binom{n}{k} \binom{n-k}{p-k} \binom{n-p}{q-k} = \binom{n}{p} \binom{n}{q}$

Published on 17 Apr 2026 - 2:49

score 134 · Answer 14 · 2026-04-15 05:42:22

134

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Using snake oil to evaluate $\sum_{k=0}^n(3k+1)\text C_n^k$

Published on 15 Apr 2026 - 5:42

score 129 · Answer 15 · 2026-04-17 14:31:55

129

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Proof of $\sum_{k=m}^{n} \binom{n}{k} \binom{k}{m} = \binom{n}{m}2^{n-m}$

Published on 17 Apr 2026 - 14:31

List Question

Bijective proof for $\sum\limits_{\substack{n_1+n_2=n\\m_1+m_2=m}}\binom{n_1+m_1}{n_1}^2\binom{n_2+m_2}{n_2}^2=\frac{1}{2}\binom{2n+2m+2}{2n+1}$?

Simplifying a double summation related to a Ramanujan-type pi formula $1/\pi$?

How to obtain explicit formula of coefficients for generating functions?

How to prove $\sum_{n\ge 0} a_n x^n = \sum_{n\ge 0} \left(\sum_{k=0}^{n}(-1)^{k+1} \binom{n}{k}a_k\right) \frac{x^n}{(x-1)^{n+1}}$?

Simplifying the sum $\sum_{ \substack{i,j,k\geq 0;\\ i+j+2k=r}} (-1)^i {n\choose i}{n \choose j}{n \choose k}$

Proof by contradiction regarding binomial coefficients and primes

Binomial identity reference request

Inverse of a Pascal-like matrix

Closed formula for $\sum_{k=0}^{\lfloor n/2\rfloor} x^k\binom{n-k}{k}$

Proving the identity $\sum_{k=0}^{m} \binom{n-k}{m-k}= \binom{n+1}{m}$ .

Alternating sum of products of half-shifted binomial coefficients

LU decomposition of a specific Vandermonde matrix

Proving the identity $ \sum_{k=0}^{\min\{p,q\}} \binom{n}{k} \binom{n-k}{p-k} \binom{n-p}{q-k} = \binom{n}{p} \binom{n}{q}$

Using snake oil to evaluate $\sum_{k=0}^n(3k+1)\text C_n^k$

Proof of $\sum_{k=m}^{n} \binom{n}{k} \binom{k}{m} = \binom{n}{m}2^{n-m}$

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