Question

Can this be done? Split Pascal's triangle (without the $1$s) with a straight line into two regions of equal sums.

score 401 · Answer 1 · 2026-04-15 04:26:53

401

Views

Can this be done? Split Pascal's triangle (without the $1$s) with a straight line into two regions of equal sums.

Published on 15 Apr 2026 - 4:26

score 74 · Answer 2 · 2026-04-15 22:13:54

74

Views

How to prove that $\big(\operatorname{sinc}(x) + \operatorname{sinc}(x-1) \big)^{*r} = \frac{\Gamma{(r+1)}}{\Gamma{(x+1)}\Gamma{(r-x+1)}}$

Published on 15 Apr 2026 - 22:13

score 148 · Answer 3 · 2026-04-14 10:19:20

148

Views

Why do these matrices within Pascal's triangle all have determinant $1$?

Published on 14 Apr 2026 - 10:19

score 93 · Answer 4 · 2026-04-12 07:16:59

93

Views

Gaussian approximation of collision time

Published on 12 Apr 2026 - 7:16

score 154 · Answer 5 · 2026-04-12 02:46:53

154

Views

Inequality of inclusion-exclusion terms

Published on 12 Apr 2026 - 2:46

score 110 · Answer 6 · 2026-04-15 04:58:44

110

Views

Looking for a function to solve for $k$ in a binomial coefficient. $n \choose k$

Published on 15 Apr 2026 - 4:58

score 26 · Answer 7 · 2026-04-15 17:59:05

26

Views

Sum of $\binom ni$, where $i$ goes from $0$ to $k < n$

Published on 15 Apr 2026 - 17:59

score 88 · Answer 8 · 2026-04-17 08:34:38

88

Views

How to prove $\sum^{m}_{k=0}(-2)^k\binom{2n-k}{n}\binom{m}{k}=0$ for $m=1,3,5,\dots,2n-1$?

Published on 17 Apr 2026 - 8:34

score 81 · Answer 9 · 2026-04-16 04:36:49

81

Views

Prove $\sum\limits_{k=0}^n \frac{(C_n^k)^2}{k+1} = \frac{(2(n+1))!}{2(n+1)^3(n!)^2}$

Published on 16 Apr 2026 - 4:36

score 410 · Answer 10 · 2026-04-11 05:39:12

410

Views

Take a random walk on Pascal's triangle, without revisits: Does the final number have infinite expectation?

Published on 11 Apr 2026 - 5:39

score 64 · Answer 11 · 2026-04-13 22:34:40

64

Views

Combinatorial interpretation of the multinomial coefficient as a product of binomial coefficients

Published on 13 Apr 2026 - 22:34

score 187 · Answer 12 · 2026-04-14 23:42:19

187

Views

Proving $\sum_{i=0}^{n}\sum_{j=0}^{n}\binom{i+j}{i}\binom{n-i}{j}\binom{n-j}{i}\ =\ \sum_{k=0}^{n}\binom{2k}{k} $

Published on 14 Apr 2026 - 23:42

score 133 · Answer 13 · 2026-04-16 11:14:47

133

Views

Conjecture: $\prod\limits_{k=0}^{n}\binom{2n}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2k}{k}$ only if $n=1,2,5$.

Published on 16 Apr 2026 - 11:14

score 110 · Answer 14 · 2026-04-17 08:06:44

110

Views

Prove that $\sum_{i=0}^{\lfloor n/2 \rfloor }\sum_{j=0}^{\lfloor n/2 \rfloor-i}{n \choose 2i}{n-2i \choose 2j}2^{n-2i-2j}=4^{n-1}+2^{n-1}$

Published on 17 Apr 2026 - 8:06

score 35 · Answer 15 · 2026-04-12 07:08:09

35

Views

Finding maximum value using expected profit with Binomially distributed demand

Published on 12 Apr 2026 - 7:08

List Question

Can this be done? Split Pascal's triangle (without the $1$s) with a straight line into two regions of equal sums.

How to prove that $\big(\operatorname{sinc}(x) + \operatorname{sinc}(x-1) \big)^{*r} = \frac{\Gamma{(r+1)}}{\Gamma{(x+1)}\Gamma{(r-x+1)}}$

Why do these matrices within Pascal's triangle all have determinant $1$?

Gaussian approximation of collision time

Inequality of inclusion-exclusion terms

Looking for a function to solve for $k$ in a binomial coefficient. $n \choose k$

Sum of $\binom ni$, where $i$ goes from $0$ to $k < n$

How to prove $\sum^{m}_{k=0}(-2)^k\binom{2n-k}{n}\binom{m}{k}=0$ for $m=1,3,5,\dots,2n-1$?

Prove $\sum\limits_{k=0}^n \frac{(C_n^k)^2}{k+1} = \frac{(2(n+1))!}{2(n+1)^3(n!)^2}$

Take a random walk on Pascal's triangle, without revisits: Does the final number have infinite expectation?

Combinatorial interpretation of the multinomial coefficient as a product of binomial coefficients

Proving $\sum_{i=0}^{n}\sum_{j=0}^{n}\binom{i+j}{i}\binom{n-i}{j}\binom{n-j}{i}\ =\ \sum_{k=0}^{n}\binom{2k}{k} $

Conjecture: $\prod\limits_{k=0}^{n}\binom{2n}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2k}{k}$ only if $n=1,2,5$.

Prove that $\sum_{i=0}^{\lfloor n/2 \rfloor }\sum_{j=0}^{\lfloor n/2 \rfloor-i}{n \choose 2i}{n-2i \choose 2j}2^{n-2i-2j}=4^{n-1}+2^{n-1}$

Finding maximum value using expected profit with Binomially distributed demand

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