If sequences $\{a_k\}$ and $\{b_k\}$ are related by $a_k = \sum_{r=0}^k \binom{k}{r} b_r$, can I compute $b_k$s in $O(1)$ time?
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
31766
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
118674
Simple upper bound for $\binom{n}{k}$
145048
Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
83638
Suppose $0,0,1,0,0,1,0,0,1,...$ is a sequence. What is the general term of this sequence?
117761
When is $\binom{n}{k}$ divisible by $n$?
21000
Calculating $\sum_{k=1}^nk(k!)$ combinatorially
79124
Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
58187
Solve the sequences inequality
119779
Sequences of arithmetic and geometric mean
110441
Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
19250
Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s
70128
Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
116491
Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$
69867
For each $\beta \in(0,1)$, give a sequence $\{a_j\}\subset(0,1)$ with $\prod (1-\alpha_j) = \beta $