All Questions
8 questions
2
votes
1
answer
113
views
Modulo $2$ binomial transform of A243499 applied $k$ times
Let $m \geqslant 1$ be a fixed integer.
Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
...
- 4,938
2
votes
1
answer
871
views
Find closed-form expression to $f(n)$
For all $n \in \mathbb{N}$, let ${\mathcal A}_n := \left\{\lceil n/2\rceil, \lceil n/2\rceil+1,\dots, n-1 \right\}$ and
$$f(n) := \begin{cases} \min\limits_{a \in {\mathcal A}_n} \frac 1 4 \binom n a ...
- 265
1
vote
0
answers
73
views
Alternating sum of integer coefficients of the triangles related to Eulerian numbers and binomial transforms
Let $W(n, k, m)$ be an integer coefficients defined for $n > 0, 1 \leqslant k \leqslant n, m > 0$ with $W(n,k,m)=0$ for $n \leqslant 0$ or $k \leqslant 0$ such that
$$
W(n, k, m) = (k+m-1)W(n-1,...
- 4,938
1
vote
0
answers
57
views
Inverse modulo $2$ binomial transform of generalised A284005
Let $m \geqslant 1$ be a fixed integer.
Let $\operatorname{wt}(n)$ be A000120, $1$'s-counting sequence: number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $f(n)$ be A007814, ...
- 4,938
1
vote
0
answers
156
views
Open tours by a biased rook (proof verification)
Related questions:
Number of open tours by a biased rook on a specific $f(n)\times 1$ board which end on a $k$-th cell from the right
Sum with products turned into subsequences
Combinatorial ...
- 4,938
1
vote
0
answers
236
views
Solving a recurrence (with the form of a convolution) involving binomial coefficients
While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$
\begin{array}{cccccccccc}
1 & = & K_{1}\tbinom{...
- 11
0
votes
1
answer
149
views
Modulo $2$ binomial transform of A124758
Let $f(n)$ be A153733, remove all trailing ones in binary representation of $n$. Here
\begin{align}
f(2n)& = 2n\\
f(2n+1)& = f(n)\\
\end{align}
Then we have an integer sequence given by
\begin{...
- 4,938
0
votes
2
answers
400
views
Solving a recurrence relation involving binomial coefficients
This question originates from a graph neural network architecture (see 1) in which an edge-labelled graph $G=(V,E,\eta)$ of size $|V|=n$ with $\eta:E\to \mathbb{R}^{s_0}$ is represented as a "tensor''...
- 493