Question

how to prove that $\sum_{k=1}^{n+1} \binom{n+1}{k} \sum_{\ell=0}^{k-1} \binom{n}{\ell}=2^{2n}$

score 35 · Answer 1 · 2026-04-15 00:20:33

35

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how to prove that $\sum_{k=1}^{n+1} \binom{n+1}{k} \sum_{\ell=0}^{k-1} \binom{n}{\ell}=2^{2n}$

Published on 15 Apr 2026 - 0:20

score 68 · Answer 2 · 2026-04-16 07:44:30

68

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Closed form for $\sum\limits_i (-1)^i \binom{m}{i} \sum\limits_{j \ge k+1} \binom{2k+1-m}{j-i}.$

Published on 16 Apr 2026 - 7:44

score 95 · Answer 3 · 2026-04-11 23:18:27

95

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Find the values of $\sum_{k=0}^{n} \frac{{n\choose k }}{k+1}$

Published on 11 Apr 2026 - 23:18

score 101 · Answer 4 · 2026-03-26 04:49:46

101

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A certain sum of multinomial coefficients

Published on 26 Mar 2026 - 4:49

score 599 · Answer 5 · 2026-04-14 14:14:57

599

Views

Number of ways to invest $\$20,000$ in units of $\$1000$ if not all the money need be spent

Published on 14 Apr 2026 - 14:14

score 1.4k · Answer 6 · 2026-04-12 13:32:18

1.4k

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Is this a new discovery in the diagonals of Pascal's triangle?

Published on 12 Apr 2026 - 13:32

score 142 · Answer 7 · 2026-03-25 06:50:39

142

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Positivity of a matrix built on Pascal's triangle

Published on 25 Mar 2026 - 6:50

score 69 · Answer 8 · 2026-04-13 03:25:13

69

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Is there a closed form of $\sum\limits_{r=0}^n \binom{n-r}{r}x^r$?

Published on 13 Apr 2026 - 3:25

score 139 · Answer 9 · 2026-04-13 16:53:15

139

Views

Asymptotic Stirling approximation of binomial coefficients

Published on 13 Apr 2026 - 16:53

score 143 · Answer 10 · 2026-04-14 16:06:42

143

Views

Find the coefficients of product

Published on 14 Apr 2026 - 16:06

score 40 · Answer 11 · 2026-04-10 17:03:26

40

Views

Proof of entity (5.24) of Concrete mathematics - Page 169?

Published on 10 Apr 2026 - 17:03

score 136 · Answer 12 · 2026-04-14 00:38:02

136

Views

Chebyshev polynomials to hypergeometric function?

Published on 14 Apr 2026 - 0:38

score 164 · Answer 13 · 2026-04-21 13:38:30

164

Views

Sum of $n$ degree derivatives taken w.r.t. different combinations of arguments

Published on 21 Apr 2026 - 13:38

score 292 · Answer 14 · 2026-04-13 23:05:17

292

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Summation of series involving binomial coefficients : $\sum_{r=1}^n {n\choose r}r^{n-r}$.

Published on 13 Apr 2026 - 23:05

score 102 · Answer 15 · 2026-04-17 14:40:26

102

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Proving that for all $1 \leq k \leq p - 1$, $p$ prime, $\binom{p}{k} \equiv 0 \bmod p$

Published on 17 Apr 2026 - 14:40

List Question

how to prove that $\sum_{k=1}^{n+1} \binom{n+1}{k} \sum_{\ell=0}^{k-1} \binom{n}{\ell}=2^{2n}$

Closed form for $\sum\limits_i (-1)^i \binom{m}{i} \sum\limits_{j \ge k+1} \binom{2k+1-m}{j-i}.$

Find the values of $\sum_{k=0}^{n} \frac{{n\choose k }}{k+1}$

A certain sum of multinomial coefficients

Number of ways to invest $\$20,000$ in units of $\$1000$ if not all the money need be spent

Is this a new discovery in the diagonals of Pascal's triangle?

Positivity of a matrix built on Pascal's triangle

Is there a closed form of $\sum\limits_{r=0}^n \binom{n-r}{r}x^r$?

Asymptotic Stirling approximation of binomial coefficients

Find the coefficients of product

Proof of entity (5.24) of Concrete mathematics - Page 169?

Chebyshev polynomials to hypergeometric function?

Sum of $n$ degree derivatives taken w.r.t. different combinations of arguments

Summation of series involving binomial coefficients : $\sum_{r=1}^n {n\choose r}r^{n-r}$.

Proving that for all $1 \leq k \leq p - 1$, $p$ prime, $\binom{p}{k} \equiv 0 \bmod p$

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