Questions tagged [sequences-and-series]

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Simplification of summation and reverse search

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $b(n)$ be an integer sequence such that $$b(2^m(2n+1))=\sum\limits_{k=0}^{m}f(m-k)b(2^kn), b(0)=1$$ Let $s(n,m)$ be an integer ...
Notamathematician's user avatar
6 votes
1 answer
251 views

On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
Tito Piezas III's user avatar
10 votes
2 answers
649 views

On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

I. Some functions As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$ $$\beta(s) = \sum_{n=1}^\infty\...
Tito Piezas III's user avatar
1 vote
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Sequences that sum up to $\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$

Let $a(n,m)$ be an integer sequence such that $$a(n,m)=\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$$ Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ $$f(n)=n-2^{\ell(n)}$$...
Notamathematician's user avatar
2 votes
2 answers
352 views

Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$

(Note: This third method continues from this post.) There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...
Tito Piezas III's user avatar
3 votes
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Ramanujan's pi formulas with a twist (nine years later)

(Note: The second method described here continues this post.) About nine years ago, I made an MO post "Ramanujan's pi formulas with a twist". An answer was informative, but not completely ...
Tito Piezas III's user avatar
4 votes
1 answer
188 views

Transformations of Ramanujan's 1/pi formulas $\sum_{n=0}^{\infty} s(n)\frac{An+ B}{C^n}$ and Monster moonshine functions

Someone with many papers on Ramanujan's work asked me how I managed to find the closed-forms for the binomial sums of level $10$ in a recent MO post. (A colleague of his wasn't able to find them.) I ...
Tito Piezas III's user avatar
6 votes
2 answers
506 views

Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped

I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
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$\frac{m(m+k+1)^n+k}{m+k}$ as closed form for subsequence of the partial sums

Let $a(n,m,k)$ be an integer sequence such that $$a(n,m,k)=\frac{m(m+k+1)^n+k}{m+k}$$ There are many sequences in the OEIS that are special cases of a given sequence family: $a(n,1,1)$ - A007051 $a(n,...
Notamathematician's user avatar
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Does this recurent matrix sequence admit an explicit writing?

I have sequence defined by : 𝐏(n+1)=(𝐈−(Ф.𝐏(n).Ф′+𝐐).𝐇′.(𝐇.(Ф.𝐏(n).Ф′+𝐐).𝐇′+𝐑)^(−𝟏).𝐇).(Ф.𝐏(n).Ф′ +𝐐) Where : P(n), Q, R are square, NxN, symmetric, positive semidefinite. R is square, ...
Poxknot's user avatar
5 votes
3 answers
592 views

On level $10$ of the McKay-Thompson series of the Monster

(For brevity, the level-6 functions have been migrated to another post.) I. Level-10 functions Given the Dedekind eta function $\eta(\tau)$. To recall, for level-6, $$j_{6A} = \left(\sqrt{j_{6B}} + \...
Tito Piezas III's user avatar
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Sum power series not continuous unit circle

This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there. Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
Libli's user avatar
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A uniform distribution problem coming from higher dimensions

Thinking about an approximation problem related to random walks, the following question came up. Suppose we have $m$ numbers $a_1, \ldots, a_m \in \mathbb{R}$ and that $b \in \mathbb{R}$ is not in the ...
Zestylemonzi's user avatar
1 vote
0 answers
46 views

A problem on monotonicity rule for the ratio of two Maclaurin power series

In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow. Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and the power series ...
qifeng618's user avatar
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Finding a distance so that this function is a contraction mapping

Let $f(x,y)=(y,\frac{2}{x+y})$ defined on $(0,\infty)\times (0,\infty)$. Is there a distance $d$ on $(0,\infty)\times (0,\infty)$ such that $f$ is a contraction of the metric space $((0,\infty)\times (...
J.Mayol's user avatar
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2 votes
1 answer
172 views

2D lattice sum with numerator

I've been struggling a bit with a double sum that arose as the trace of an operator: $$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$ where $n$ is an even natural number. Is there ...
R Grady's user avatar
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Product as closed form for subsequence of the partial sums

Let $a(n,m,k)$ be an integer sequence such that $$a(n,m,k)=\prod\limits_{q=0}^{n-1}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{m-i-1}\binom{i+j-1}{j}k^{i+j}q^i$$ Let $$\ell(n,m)=\left\lfloor\log_m n\...
Notamathematician's user avatar
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On a generalization of A113227 as a subsequence of the partial sums

This question is just a generalization of the one of my previous questions. Let $$a(n,m,k)=\sum\limits_{i=1}^{n}u(n,m,k,i)$$ where $$u(n,m,k,i)=u(n-1,m,k,i-1)+(m-1)(i+k-1)\sum\limits_{j=i}^{n-1}u(n-1,...
Notamathematician's user avatar
5 votes
3 answers
2k views

How many digits of $\sqrt{2}$ are known to date?

How many digits of $\sqrt{2}$ are known to date, in base 10 and in base 2? I am trying to produce the largest sequence known to date, and would like to sense if I can do it either alone or with hiring ...
Vincent Granville's user avatar
4 votes
0 answers
85 views

Closed form for subsequence of the partial sums of generalized A329369

Let $a(n,m,k)$ be an integer sequence such that $$a(n,m,k)=\sum\limits_{i=0}^{n}{n\brace i}(m-k)^{n-i}\prod\limits_{j=0}^{i-1}(kj+1)$$ Here ${n\brace k}$ is the Stirling number of the second kind. ...
Notamathematician's user avatar
3 votes
1 answer
130 views

Sequences that sum up to Dowling numbers

Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see A007405 and its CROSSREFS section) with e.g.f. $$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$ ...
Notamathematician's user avatar
2 votes
1 answer
148 views

The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$

Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of $$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$ as $\lambda\to 0^{+}$ and as $\lambda \...
Medo's user avatar
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0 answers
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Product-based binary numeration system

I am looking at the following binary numeration system: $$x =\prod_{k=1}^\infty \Bigg(1+\frac{d_k(x))}{2^k}\Bigg), \quad d_k(x)\in \{0, 1\}.$$ The $d_k$'s are the digits, and $x$ is between $1$ (all ...
Vincent Granville's user avatar
1 vote
0 answers
107 views

Value of $\pi$ and algorithm for Bernoulli numbers

Chowla and Hartung provide an "algorithm" for computing Bernoulli numbers in this paper. In particular, if the Bernoulli numbers are defined by $$\frac{x}{e^x-1}=1-\frac{x}{2}+\sum_{n=1}^\...
japjap's user avatar
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2 votes
0 answers
102 views

Sequences that sum up to the many sequences in the OEIS

Let $$a(n,m,k)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj\geqslant 0]\binom{n}{j}\binom{n+kj}{j-1}(m-1)^{j-1}$$ Here square brackets denote Iverson brackets. There are many sequences in the OEIS that are ...
Notamathematician's user avatar
2 votes
2 answers
219 views

Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$

When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
Caleb Briggs's user avatar
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2 votes
0 answers
59 views

Factor group of all the sequences by the subgroup of bounded sequences

Consider the group G of the sequences of real numbers (the group operation is addition). It contains a subgroup H of bounded sequences. Is there any nice description of the factor group G/H ? It is ...
Nikita Kalinin's user avatar
6 votes
1 answer
254 views

Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$

Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$. The sequence begins with $$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...
Notamathematician's user avatar
0 votes
1 answer
129 views

Does rapid convergence of the Cesaro sums imply convergence of the original sequence?

Question: Let $a_n$ be a sequence of real numbers. Is it true that for every $\varepsilon > 0$, if $$\left \lvert \frac{1}{N} \left ( \sum_{n=0}^{N-1} a_n \right )\right \rvert < \frac{1}{N^{1+\...
Nate River's user avatar
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3 votes
1 answer
251 views

Where does the Weierstrass expansion of $\operatorname{sn}$ come from?

In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $\operatorname{sn}$') $$\operatorname{sn}u=\frac{B}{A}$$ where $...
japjap's user avatar
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2 votes
0 answers
67 views

Integer coefficients such that $T(n,k)=R(n,k)-R(n,k-1)$

Let $a(n)$ be A000085, i.e., the number of self-inverse permutations on $n$ letters, also known as involutions; number of standard Young tableaux with $n$ cells. Here $$a(n) = a(n-1) + (n-1)a(n-2), a(...
Notamathematician's user avatar
7 votes
1 answer
302 views

If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$

I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
Caleb Briggs's user avatar
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1 vote
0 answers
53 views

Recurrence for the number of permutations with a given excedance set

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ $$f(n)=n-2^{\ell(n)}$$ $$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$ $$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...
Notamathematician's user avatar
1 vote
0 answers
132 views

Recurrence for the A284005

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ $$f(n)=n-2^{\ell(n)}$$ $$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$ $$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...
Notamathematician's user avatar
0 votes
0 answers
68 views

Permutation that produces permutations

Let $f(n)$ be A000045, i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$. Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with $$1, 2, 3, ...
Notamathematician's user avatar
0 votes
1 answer
124 views

proving inequality in Riemann zeta function

Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}. $$ When this ...
MrPie 's user avatar
  • 185
2 votes
0 answers
76 views

Uniqueness of the permutation

Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$. Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with $$1, 2, ...
Notamathematician's user avatar
5 votes
2 answers
216 views

Continuous functions on $[0,1]^\omega$ and a product lower bound

I have a concrete question about continuous functions on $X = [0,1]^\omega$ (with the product topology). The map $f:X\to [0, 1]$ given by $(x_i)\mapsto \prod x_i$ is well-defined and Borel but not ...
dnkywin's user avatar
  • 53
1 vote
0 answers
164 views

Ask for a proof of an inequality involving the Bernoulli numbers

Let $B_k$ be the Bernoulli numbers and let \begin{equation} T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1. \end{equation} Prove the inequality \begin{equation*} \frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
qifeng618's user avatar
  • 836
7 votes
3 answers
581 views

Is the sequence $\{n/\{ \sqrt{2}n\} \}$ uniformly distributed in $[0,1)$?

A classical theorem of Kronecker says that the sequence $(\{\alpha_1 n\}, \{\alpha_2 n\},\dots,\{\alpha_d n\})$ ($n \in \mathbb{N}$) is uniformly distributed in $[0,1)^d$ provided that $1,\alpha_1,\...
Jakub Konieczny's user avatar
4 votes
4 answers
635 views

What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?

It is known that \begin{equation*} \tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2} \end{equation*} and \begin{equation*} \ln\tan x=\ln x+\...
qifeng618's user avatar
  • 836
5 votes
3 answers
1k views

Solving a limit about sum of series

what's the limit of $\sqrt{1-t}\sum _{n=0}^{\infty}t^{n^2}$ as $t$ goes to the left of $1$? i.e. $t\to 1^{-}$? I tried several times but failed. Here is my thought: This is a $0\cdot\infty$ problem, ...
Xu Shan's user avatar
  • 95
1 vote
0 answers
109 views

Existence of binary permutations with a given property

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$f(n)=n-2^{\ell(n)}$$ Let $a(n)$ be a permutation of the nonnegative integers such that $a(0)=0$, $a(n)=n$ if $n$ is a power of $2$ and ...
Notamathematician's user avatar
9 votes
2 answers
402 views

How to prove this sum involving powers of cosec is an integer?

It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$. $F(m,N)=\frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\...
MilesB's user avatar
  • 201
2 votes
1 answer
151 views

Permutation and its binary analog

Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$. Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with $$1, 2, ...
Notamathematician's user avatar
1 vote
0 answers
88 views

limsup of sequence

Let $\mathbb{Z}_{\geq 0}[|t|]$ be the ring of power series with non-negative integer coefficients and consider the power series $$P(t) = \sum_{i=0}^ \infty a_i t^i \in \mathbb{Z}_{\geq 0}[|t|]$$ $$P^2(...
SKS's user avatar
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1 vote
0 answers
80 views

Infiniteness of the pairs of sequences with a given conditions

Let $$\varphi=\frac{1+\sqrt{5}}{2}$$ Let $$a_1(n)=\left\lfloor n\varphi \right\rfloor, a_2(n)=n+a_1(n)$$ Let $\operatorname{tr}(n)$ be A007814, i.e., the number of trailing zeros in the binary ...
Notamathematician's user avatar
6 votes
0 answers
168 views

Computing residues at $\infty$

As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
Caleb Briggs's user avatar
  • 1,662
0 votes
0 answers
49 views

Stolarsky array and Stolarsky representation

Let $T(n,k)$ be A035506, i.e., Stolarsky array read by antidiagonals. Here we consider that $T(n,k)=0$ for $n<1, k<1$. Let $a(n)$ be A200714, i.e., Stolarsky representation interpreted as binary ...
Notamathematician's user avatar
4 votes
2 answers
394 views

The set of all possible values of subseries of a convergent positive term series

Inspired by The set of all limits of sub-series of an absolute convergent series is the following true?: Let $a_n$ be a strictly decreasing sequence and $\sum_1^\infty a_n=\ell<\infty$ is a ...
Ali Taghavi's user avatar

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