Questions tagged [sequences-and-series]
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.
1,713
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A discrete version of Poincaré's inequality
Given a (bounded) sequence $\{q_n\}_{n\geq 0}$ such that $\lvert q_n\rvert \leq 1$ for all $n \geq 0$ and $\sum_{n\geq 0} q_n = 0$. We can impose the condition that $\sum_{n\geq 0} \lvert q_n\rvert \...
2
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1
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988
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Switching order of limits in double sequences
By trying to extend certain limit properties of sequences from compact subsets to the entire set, I cam up with something that can be formed in the following question.
Let $a_{mn}$ be a double ...
2
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1
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322
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Simplify multiple sum involving rising factorials
(Previously asked in MSE, no answer even with bounty offer)
In the course of a calculation, I arrived at the quantity
$$ f(x,y,a,b)= \sum_{n,m,i,r,q,l\ge 0}\sum_{k=0}^{n+m} K_{n,m,i,r,l,q,k}\frac{(x)^{...
1
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1
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107
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Can we write $e^{-\alpha x}$ as $\sum_{n=0}^\infty c_n\left(\alpha\right)\gamma\left(x\right)^n$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$
Do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the following equation is true ...
3
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0
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221
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Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series
Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...
3
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1
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202
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Is normalcy preserved under the swapping operation?
Let $\mathbb{N}$ denote the set of non-negative integers. We say that a sequence $f:\mathbb{N}\to \{0,1\}$ is normal if every finite $\{0,1\}$-sequence appears in $f$.
Let the swapping operation $\...
2
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1
answer
211
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Identity involving double sum with binomials
(asked previously in MSE here)
In the course of a calculation, I have met the following complicated identity. Let $A$ and $a$ be positive integers. Then I believe that
$$ \sum_{B\ge A,b\ge a} (-1)^{a+...
2
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1
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374
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A conjecture relating an integral and a sum, the floor function and squares
I've found through evidence and have conjectured on a math publication that:
$$\Big\lfloor\int_1^\infty (k^{1/(k^{1+1/\sqrt{x}})} - 1)dk\Big\rfloor = \Big\lfloor\sum_{k=1}^{\infty}k^{1/(k^{1+1/\sqrt{x}...
2
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1
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286
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Convergence of $\sum(n^p\sin^qn)^{-1}$
I've been recently interested in the problem of convergence of the function in such form: $\displaystyle \sum_{n=1}^\infty\frac1{n^p\sin^q n}$.
I saw there's been discussion here when $p=3, q=2$ and $...
0
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1
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72
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Proof of yet another extension of deterministic variant of "(Almost) Supermartingale" convergence theorem
In this question, there is a proof for deterministic version of "Almost Supermartingale"
Question: Can we extend [1] as following? If yes, can we prove it?
Let the non-negative sequences be ...
1
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0
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44
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Permutation of nonnegative integers applied to the numbers $n$ whose binary expansion does not begin with $11$
Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and
$$n=2^{b_1}(1+2^{b_2+1}(1+2^{b_3+1}(1+\cdots(1+2^{b_{\operatorname{wt}(n)-1}+1}(1+2^{...
6
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1
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216
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Sequence A76132 eventually periodic modulo $2,3$ and $5$
Sequence A76132 starting as $1,1,2,4,10,36,218,\ldots$ of the OEIS is recursively defined by $a(1)=1$
and $a(n)=\sum_{k=1}^{n-1}a(n-k)^k$ for $n\geq 2$.
It is eventually periodic of period 1,1 and 34 ...
1
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0
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143
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Pairs of functions with $\sum_{n} (f \circ g)(n) = \sum_{n} (g \circ f)(n) $
I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with ...
-1
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1
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187
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Is the unordered sum of measurable functions measurable?
Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
5
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1
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177
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A common combinatorial description for a certain type of recurrences
For integer-valued sequences $(x_n)_{n=0}^\infty$, consider recurrences of the form
$$x_n=ax_{n-1}+(bn+c)x_{n-2} \tag{$*$}\label{star}$$
for $n\ge2$, where $a,b,c$ are integers.
There seem to be many ...
10
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1
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366
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Looking for a "clever" argument for a $q$-series identity
Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation.
$$\prod_{k\geq1}...
2
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0
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154
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The recurrence relation $x_{n+1}=x_{n}+x_{n-1}+c\,(x_{n},\,x_{n-1})$
I'm trying to study the behaviour of the recurrence relation
$$x_{n+1}=x_{n}+x_{n-1}+c(x_{n},x_{n-1})\;\;\;\;\;\;\;c,x_{n} \in \mathbb Z$$
where $(x_{n},x_{n-1})$ is the gcd of $x_{n}$ and $x_{n-1}$.
...
1
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0
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73
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Sum of changing termes serie [closed]
I am looking towards which value tends the sum of the inverses of the odd numbers going from 1 / (2k + 1) to 1 / (4k-1) when k tends towards + infinity :
for k = 1 there is just 1/3
for k = 2 it is ...
7
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3
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975
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Question on OEIS A000085
The OEIS sequence A000085 is defined by
$$ a_n \!=\! (n-1)a_{n-2} + a_{n-1} \;\text{with }\; a_0\!=\!1, a_1\!=\!1.$$
If $n$ of the form $b^2-b+1, b \in \mathbb{N}, b > 2, \;\text{then: }\;$ $$ \...
7
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2
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465
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Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$
Let $f:\mathbb{R}\to\mathbb{R}$ be the function $$f(x) = \sum_{k=1}^{\infty} \sqrt{\max\left\{1 -\frac{k^2}{x^2},0\right\}}.$$
Numerical experiments suggests that there exists $n\in\mathbb{N}$ and a $...
1
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1
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171
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Relation between $\sum_{k\ge 0}\binom {n+k}{k}a_k $ and $\sum_{k\ge 0}\binom {n+k}{k}\frac{a_k}{k}$
Let $\{a_k\}(k\ge 0)$ be a sequence of nonzero real numbers which changes signs infinitely often. Suppose $|a_k|\to 0 $ and $|a_k|$ decreases fast. Let $n$ be a positive integer. What's the relation ...
5
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3
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296
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Closed formula for $(-1)$-Baxter sequences
The number of the so-called Baxter permutations of length $n$ is computed by
$$a_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$
There has also been a ...
2
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1
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291
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Difference between $n$-th and $(n-1)$-th composite numbers
Let $f(n)$ = 1 if $n$ belongs to A014689, $\operatorname{prime}(n)-n$, the number of nonprimes less than $\operatorname{prime}(n)$. Here $\operatorname{prime}(n)$ is the $n$-th prime number, $\...
1
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0
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58
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A lower bound for a sum related to the $j$-invariant function
There are some days that I am thinking in the following problem.
For any positive integer $x$, let $t(x)$ be a real number which a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \...
3
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3
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442
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Growth of the coefficients of the inversion of the $j$-invariant function
We have the $j$-invariant defined as
I have that
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.
The inversion ...
6
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0
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196
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Where to cut off a double sum?
I have to compute a double infinite sum to within a given accuracy $\epsilon$. Let us say the sum is of the form
$$\sum_{m\geq 1} \sum_{n\geq 1} \frac{a_{m,n}}{m^2 n^2 \max(m,n)},$$
where $|a_{m,n}|\...
6
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2
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363
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Sequence of $k^2$ and $2k^2$ ordered in ascending order
Let $\eta(n)$ be A006337, an "eta-sequence" defined as follows:
$$\eta(n)=\left\lfloor(n+1)\sqrt{2}\right\rfloor-\left\lfloor n\sqrt{2}\right\rfloor$$
Sequence begins
$$1, 2, 1, 2, 1, 1, 2, ...
1
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1
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101
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Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function
Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series
$$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
2
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1
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Are there any published studies on cases of infinite sums for which the Euler–Maclaurin summation method yields the exact evaluation?
The Euler–Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(...
1
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1
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433
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Closed form series for reciprocal cubic function
consider a cubic of the form f(x)=$x^3-2x+z$
Is it possible to derive a power series of coefficients for the function $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno'...
7
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1
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1k
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$\sum a_n = 0$ but $\sum \frac{a_n}{n} = \infty$
I'm hoping to find an explicit construction for a sequence such that $\sum a_n = 0$ but $\sum \frac{a_n}{n} = \infty$, or a proof that one cannot exist. So far, I have a good idea of how we can ...
13
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0
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299
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Convergence of the series $\sum_{n=1}^\infty \frac{(2+\sin n)^n}{3^n n^a}$ for $a\in(0,1)$
This is inspired by this Math.SE question, for $a=1$.
Borwein, Bailey, and Girgensohn pose in their book ([1,Problem 35]) as an open problem the convergence of the series
$$\sum_{n=1}^\infty \frac{(2+\...
2
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0
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213
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Two conjectures about generalised A329369
Let $m \geqslant 2$ be a fixed integer.
Let
$$\operatorname{wt}(n,m)=\operatorname{wt}\left(\left\lfloor\frac{n}{m}\right\rfloor,m\right)+n\bmod m, \operatorname{wt}(0,m)=0$$
Then we have an integer ...
5
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1
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655
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Is this infinite product entire?
Let $(z_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum_{i=1}^{\infty} \vert z_i \vert = \infty$,$\sum_{i=1}^{\infty} \vert z_i \vert^2 < \infty$ and $...
4
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1
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298
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Taylor coefficients of Hadamard product
I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$
where $E_1(z):=(1-z)e^z$ is the first elementary ...
3
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2
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Does my construction always result in a stationary Poisson point process of intensity $1$? How so?
My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a ...
2
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1
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110
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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 $n$.
...
0
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1
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149
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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{...
1
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0
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56
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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, ...
8
votes
0
answers
513
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Concave and other bounded functions: Series representation and converging polynomials
Main Question
Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
1
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0
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154
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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 ...
2
votes
2
answers
170
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Modulo $2$ binomial transform of $m^n$
Let $m \in \mathbb{R}$.
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 $n$.
Let $g(n)$ be ...
5
votes
1
answer
212
views
Absolute summability of multiplication operators on $\ell_p$
A linear bounded operator $T:X\to Y$ between Banach spaces is called absolutely summing if for every unconditionally convergent series $\sum_{i\in\omega}x_i$ in $X$ the series $\sum_{i\in\omega}\|T(...
1
vote
2
answers
146
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Natural boundary with non-zero "thickness"
Many series (in fact, "most" Taylor series) have a natural boundary at the unit circle. This boundary is only 1 dimensional, in the sense that only along the unit circle is it true that ...
3
votes
0
answers
145
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Combinatorial interpretation of inverse modulo $2$ binomial transform of A284005
My question is related to the following:
Sum with products turned into subsequences
We have an identity
$$a(n, -1) = \sum\limits_{j=0}^{2^{\operatorname{wt}(n)}-1}(-1)^{\operatorname{wt}(n)-\...
9
votes
1
answer
534
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Infinite series with inverse trigonometric functions
Consider the infinite series
$$
F(s)=\sum_{k=1}^\infty \frac{1}{k^{2s+1} \sin(k \pi \sqrt{2})}
$$
Prudnikov Vol.1 , gives a result for this sum in 5.4.16.13 for $s=1.$
$$
F(1)=-\frac{13 \pi^3}{360 \...
3
votes
1
answer
290
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Sum with products turned into subsequences
Let $p, q \in \mathbb{Z}$.
Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and
$$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$...
7
votes
1
answer
690
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One conjecture by sequencedb.net
Let $a(n)$ be A214973, number of terms in greedy representation of $n$ using Fibonacci and Lucas numbers.
Let $b(n)$ be A329320, sequence which arises from attempts to simplify computing of A329319. ...
1
vote
0
answers
58
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Subsequences related with square table
Let $m\geqslant1$ be a fixed integer.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $...
2
votes
0
answers
90
views
Subsequence of Laguerre polynomials
Let $m\geqslant1$ be a fixed integer.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $...