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Results tagged with sequences-and-series
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user 46140
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
2
votes
Accepted
Closed form for $\sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]$ where $\operatorname{wt}(...
For $n > 0$ let $\ell(n) = \lfloor \log_2(n) \rfloor$ so that $n = 2^{\ell(n)} + r_n$ where $0 \le r_n < 2^{\ell(n)}$. Note that $\ell(n) = T(n, 1) - 1$.
Then by partitioning the numbers $k$ up to $n$ …
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4
votes
Accepted
Correctness of the algorithm for the A329369, A347205 and related sequences
Generalise to $$b(2^m(2k+1)) = \sum\limits_{j=0}^{m}C_{m+1,j} \, b(2^jk), \\
b(0) = 1$$
Consider the infinite matrices: $$M_0 = \begin{pmatrix} 0 & 1 & 0 & 0 & \cdots \\
0 & 0 & 1 & 0 & \cdots \\
0 & …
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2
votes
Sequence derived from transform of a given vector (with Fibonacci as partial sums)
Not a complete answer, but too long for a comment and addressing the conjecture which I take to be the most important part of the question.
The double-loop transformation process seems familiar to me …
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1
vote
An identity for the ratio of two partial Bell polynomials
Counterexample: consider $\ell = 3$, $m = 1$. The LHS is $$\frac{B_{7,3}({\color{red} 1}, {\color{green} 0}, {\color{blue} 1}, 0, 9)} {B_{5,3}({\color{red} 1}, {\color{green} 0}, {\color{blue} 1})} =
…
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4
votes
Accepted
Property of some permutations of non-negative integers such that $a(n)<2^k$ iff $n<2^k$
I'm going to use $\operatorname{msb}$ (for most significant bit) as an alias of $f$.
Since $q_i$ is a permutation, the property that $q_i(n)<2^k$ iff $n < 2^k$ is equivalent to $\operatorname{msb}(q_i …
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3
votes
Accepted
Permutation of the natural numbers from operation related to binary expansion of $n$
$\ell(0)$ is problematic, so I will assume that you actually mean to restrict to positive integers rather than natural numbers.
We can rephrase the construction of $a(n)$ to emphasise the increment: l …
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2
votes
Accepted
$q$-series and Stirling of the 1st kind
QUESTION. Is this true? Or, can you provide a reference to it.
$$\mathbf{F}_a(q)=\frac1{(2a-1)!}\mathbf{G}(\mathbf{G}^2-1^2)(\mathbf{G}^2-2^2)(\mathbf{G}^2-3^2)\cdots(\mathbf{G}^2-(a-1)^2);$$
where w …
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1
vote
Accepted
Series reversion using something like continued fraction
We assume $F(0) \neq 0$, since otherwise we don't satisfy the assumptions for the series reversion. Let $G = G(0)$ be the fixpoint of the recurrence given:
$$G(x) = F\left(\frac{x}{G(x)}\right)$$
Mult …
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3
votes
Operation preserving log-concavity of sequences
If my calculations are correct, a counterexample for question 2 is $$f= \frac{720 + 1684x + 1350x^2 + 585x^3 + 90x^4 + 11x^5}{120} \\
g = \frac{600 + 1434x + 1175x^2 + 535x^3 + 85x^4 + 11x^5}{120}$$ h …
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1
vote
Accepted
Sequences that sum up to Dowling numbers
Cleaning up the notation a bit,
$$b_{m,k}(n) = m\, b_{m,k}(n-2^{\ell(n)}) + k \sum_{j=0}^{\ell(n)-1} [n \,\&\, 2^j = 0] \,b_{m,k}(n - 2^{\ell(n)} + 2^j)$$ where $\&$ is bitwise AND.
$$s_{m,k}(n) = \su …
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1
vote
Accepted
Recurrence for the number of steps required to get one ball in each box
Generalise $a$: $a(n, k)$ is the number of steps to perform this process with $n+k$ boxes and balls starting with $n \ge 1$ balls in the first box and one ball each in the next $k$ boxes. Then the ori …
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15
votes
Accepted
Multiple roots of polynomials with coefficients $\pm 1$
Question P. Can a polynomial $P(x)=\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?
Yes. The following four Littlewood polyn …
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3
votes
$a(16n+k)=b(16n+k)-c(16n)$ for $n\geqslant0$, $0 < k < 16$ where $c(n)=b(n)-a(n)$
In this answer, $\wedge$ denotes bitwise AND and $[\;]$ are Iverson brackets.
$$b(n) = \sum_{k \ge 0} \sum_{j \ge 0} (-1)^{k}(j+1) [n \wedge 2^{j+k}] = \sum_{e \ge 0}[n \wedge 2^e] \sum_{j=0}^e (-1)^{ …
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4
votes
Accepted
Periodic sequences of integers generated by $a_{n+1}=\frac{\operatorname{rad}(pa_{n})}{p}+\f...
For any odd $p$, $q$ (not necessarily prime) the values modulo $2$ follow a cycle of order 3.
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7
votes
Accepted
Numbers $m$ for which coefficients of the polynomial $p(m,x)$ are relatively prime
Counterexample: $463 \in b(n)$ (it's a prime and $464 = 2^4 \cdot 29$ is not squarefree), but $463 \not \in a(n)$ because it's a factor of the GCD of the coefficients of $p(463, x)$.
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