# 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.

1,788
questions

2
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1
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661
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### The average of an infinite sequence

I ran into the following problem when I did scientific research.
Consider an infinite sequence $\{a_{i}\}$ for $p>u>0$. $a_{1}=0$. If $a_{i}\geq p$, $a_{i+1}=a_{i}-p$; otherwise, $a_{i+1}=a_{i}+...

2
votes

0
answers

21
views

### On doubling or addition formulas for the sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$

We are interested which integer sequences are efficiently computable
possibly over finite rings.
Define the integer sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$
with initial terms $a(0),a(1)$...

0
votes

1
answer

85
views

### Summation of binomial coefficients with alternating signs

For a fixed $\alpha > 1$ and integer $n$, I want to provide some bounds or scaling results for the following summations
$$S_1(n,\alpha) = \sum_{k = 1}^{n} {n \choose k} (-1)^{k + 1} k / (\alpha k + ...

4
votes

2
answers

281
views

+50

### Conjectured Somos-like closed form of recurrences with polynomial coefficients

From Our short paper
For polynomial $F$ with integer coefficients, define the recurrence
$f(n)=F(n,f(n-1),f(n-2),...,f(n-d))$. We conjecture that
$f(n)$ satisfy Somos like sequence
$f(n)=\frac{G(f(n-1)...

4
votes

1
answer

143
views

### Closed form expression for $\sum_{n=0}^{\infty} J_n^2(x) \cos(ny)$, where $J_n(x)$ is the Bessel function of order $n$

Anyone can find/calculate a closed form expression for the sum
$$
\sum_{n=0}^{\infty} J_n^2(x) \cos(ny),
$$
where
$J_n$ is the Bessel function?

2
votes

2
answers

246
views

### 5 different ways to define the same family of integer sequences

Let ${n \brace k}$ be a Stirling number of the second kind.
Let $A_n(x)$ be an Eulerian polynomial. Here
$$
A_n(x) = \sum_{i=0}^{n}i!{n \brace i}(x-1)^{n-i}.
$$
Let $a_1(n,p,q)$ be the family of ...

1
vote

1
answer

131
views

### An inequality about binomial distribution

Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...

0
votes

1
answer

77
views

### Do disjunctive sequences eventually get palindromic at some point?

I have a friend who is very interested in math and has been thinking about a problem involving disjunctive sequences. For his birthday, I would like to give him an answer to his question, either by ...

3
votes

0
answers

224
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### Exotic series for some mathematical constants from String Theory

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and ...

0
votes

0
answers

128
views

### Integer coefficients and continued fractions

Let $a(n,p,q)$ be the family of integer sequences such that ordinary generating functions for it are $\frac{1}{G_1(0,x)}$ where $G_1(0,x)$ are continued fractions such that
$$
G_1(j,x)=1-\cfrac{(qj+1)...

3
votes

0
answers

264
views

### How many roots does $\tan(z)-z^n$ have for $n \in \mathbb{N}$, $\frac{-\pi}{2}\le \Re(z)\le \frac{\pi}{2}$?

I asked this question on MSE here.
I am investigating the number of roots of the equation
$$\tan(z) - z^n = 0$$
within the vertical strip $|\text{Re}(z)| \leq \frac{\pi}{2}$ for positive integers $n$...

3
votes

1
answer

116
views

### $R$-recursion for unsigned Genocchi numbers (of first kind) of even index

Let $G_n$ be A036968 (i.e., Genocchi numbers). Here
$$
\frac{2t}{1+e^t}=\sum\limits_{n=0}^{\infty}G_n\frac{t^n}{n!}.
$$
Also
$$
t\tan\left(\frac{t}{2}\right)=\sum\limits_{n=1}^{\infty}(-1)^n G_{2n}\...

2
votes

1
answer

138
views

### $R$-recursion for Fibonacci numbers using signed Catalan numbers

Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1.
$$
Let $C_n$ be A000108 (i.e., Catalan numbers). Here
$$
C_n = \frac{1}{n+1}\binom{2n}{n}.
$$
Let
$...

1
vote

1
answer

54
views

### Simplest way to generate integer coefficients with row sums equal to the terms of an arbitrary given sequence

Let $f(n)$ be an arbitrary function.
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\operatorname{wt}(2n+1) = \operatorname{wt}(n) + 1, \\
\...

1
vote

0
answers

165
views

### Integer coefficients and integrals

Let $a(n,p,q)$ be the family of integer sequences such that exponential generating functions for it satisfy
$$
A_1(x)=\exp\left(x + p\int\int (A_1(x))^q \, dx \, dx\right).
$$
Let $b(n,p,q)$ be the ...

0
votes

0
answers

28
views

### Short periods modulo primes of linear recurrences with polynomial coefficients

Let $f_i(x)$ be polynomials with integer coefficients.
Define the integer linear recurrence with polynomial coefficients:
$$
a(n)=f_1(n) a(n-1)+f_2(n)a(n-2)+\cdots +f_d(n) a(n-d)
$$
and the initial ...

0
votes

0
answers

52
views

### Sequences that sum up to sums of integer coefficients

Let
$$
T(n,k,p,q,r,s) = (q(k-1)+1)T(n-1,k,p,q,r,s) + s(n+r(k-1)+p-2)T(n-1,k-1,p,q,r,s), \\
T(n,1,p,q,r,s) = 1, \\
T(n,0,p,q,r,s) = T(0,k,p,q,r,s) = 0
$$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\...

1
vote

0
answers

84
views

### Closed form for the A357990 using A329369 and generalised A373183

Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor, \\
\ell(0) = -1
$$
Let
$$
f(n) = \ell(n) - \ell(n-2^{\ell(n)}) - 1
$$
Here $f(n)$ is A290255.
Let $A(n,k)$ be a square array such that
$$
A(n,k)...

1
vote

0
answers

153
views

### Property of a sequence on $\mathbb Q[\sqrt m~]$

Given that $a_{1} = \sqrt m$ in which $m$ is a integer that is not the square of any integer. And $$a_{n+1}=\frac{[a_{n}]}{\{a_{n}\}}$$where $[~ ]$ and $\{~ \}$ respectively represent the integer part ...

7
votes

1
answer

671
views

### Closed form for $\sum_{n=0}^\infty \frac1{2^{2^n}}$?

Is the sum of series $\displaystyle \sum_{n=0}^\infty \frac1{2^{2^n}} = \frac12 + \frac14 + \frac1{16} + \frac1{256} + \frac1{65536} + \dotsb \approx 0.8164215090218931$ representable in a closed form?...

6
votes

0
answers

742
views

### For all $n\in \mathbb{N}$, How to find $\min\{ m+k\}$ such that $ \binom{m}{k}=n$?

I asked this question on MSE here.
Most numbers in pascal triangle appear only once (excluding the duplicates in the same row of the Pascal's triangle) but certain numbers appear multiple times. ...

1
vote

0
answers

171
views

### Solution of recurrence relation with summation

I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...

3
votes

0
answers

114
views

### Sequence that sums up to A014307

Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $n \brace k$ be a Stirling number of the second kind.
Let $a(n)$ be A014307. Here
$$
A(x) = \sum\limits_{k=0}^{\infty} \frac{a(k)}{...

0
votes

0
answers

121
views

### How to prove the convergence of the following series involving Gamma function?

Consider the following result（$d$ denotes the dimensions and $0<t<T$）
$$c\left(\sum_{j=0}^\infty\frac{\Gamma^j(1-\kappa)}{\Gamma((j+1)(1-\kappa))}t^{j(1-\kappa)-\kappa}\right)^{\frac{1}{2}}\leq ...

5
votes

1
answer

259
views

### Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?

Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...

2
votes

1
answer

165
views

### Simplification of the closed form for the A329369

Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let ${n \brace k}$ be a Stirling number of the second kind.
Let
$$
f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace ...

2
votes

1
answer

114
views

### Recursion for the sum with Stirling numbers of both kinds

Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $n \brace k$ be a Stirling number of the second kind.
Let
$$
f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace j}...

2
votes

0
answers

251
views

### Gosperable formulas

Below we give examples of Gosperable formulas
\begin{align}
& (1) \quad \sum_{n=0}^{\infty} \frac{(\frac12)_n^6}{(1)_n^6} \, \frac{1-12n^2+48n^4}{(1-2n)^3} = \frac{8}{\pi^3}, \\
& (2) \quad \...

10
votes

1
answer

1k
views

### Duplicating Matryoshka dolls

We start with a single doll of size $1$. Every second, independently of each other, every doll present produces a new doll of half its size with probability $\frac{1}{2}$. What is the expected size of ...

2
votes

0
answers

97
views

### Another (unique) algorithm for the A329369

Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \leqslant k ...

29
votes

1
answer

3k
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### Proof of "Possible new series for $\pi$" without use of physics

Related post: The post Possible new series for $\pi$ is about whether the identity is new, so to avoid confusion I was advised to ask this question separately.
I am looking for a proof of the ...

9
votes

1
answer

493
views

### Does the sequence formed by Intersecting angle bisector in a pentagon converge?

I asked this question on MSE here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...

1
vote

0
answers

80
views

### Closed form for the family of polynomials

Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $R(n,x)$ be the family of polynomials such that
$$
R(2n+1,x) = xR(n,x), \\
R(2n,x) = x(R(n,x+1) - R(n, x)), \\
R(0, x) = x
$$
Let $\...

65
votes

8
answers

7k
views

### Possible new series for $\pi$

In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \...

1
vote

0
answers

54
views

### Simple recursion for the A329369 using Stirling numbers of both kinds

Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $n \brace k$ be a Stirling number of the second kind.
Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with ...

0
votes

0
answers

38
views

### Mellin transform of confluent Lauricella hypergeometric function

The $F_D^{(n)} $ Lauricella's hypergeometric function can be defined as follow
$$F_D\left(a,b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(a\right)_{m_1+\cdots+...

1
vote

0
answers

131
views

### Sequence that sums up to A000153

Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \leqslant k ...

3
votes

0
answers

86
views

### Recursion for reversed rows of the A373183 using unsigned Stirling numbers of the first kind

Let $\left[{n \atop k}\right]$ be unsigned Stirling numbers of the first kind. Here
$$
\left[{n \atop k}\right] = (n-1)\left[{n-1 \atop k}\right] + \left[{n-1 \atop k-1}\right], \\
\left[{n \atop 0}\...

16
votes

3
answers

4k
views

### Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?

In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...

3
votes

1
answer

117
views

### Counting equal covering sets

Definition. We call a set $C$ of sets to be an equal covering set of $S$ if the elements of $C$ are all the same size and each element of $S$ is contained an equal number of times throughout the sets ...

1
vote

0
answers

84
views

### Simpler recursion for the A358612

Let $T(n,k)$ be an integer coefficients (A358612) such that
$$
T(2n+1, k) = kT(n, k) + T(n, k-1), \\
T(2n, k) = kT(n, k) + T(n, k-1) - \frac{T(2n, k-1) + T(n, k-1)}{k-1}, \\
T(n, 1) = T(0, 2) = 1
$$
...

0
votes

0
answers

96
views

### integral of exponential of Fourier series

I have encountered the following integral:
\begin{equation}
\int_0^{1} e^{-i F(x)} dx, \quad F(x) = \sum_{k=1}^L a_k \sin(2\pi k x) + b_k \cos(2\pi k x).
\end{equation} I have found several great ...

1
vote

0
answers

110
views

### Representing A329369 using A358612

Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...

1
vote

1
answer

180
views

### Does any such family of functions exist?

Is there a sequence of non-zero bounded smooth functions $f_1,f_2,\ldots,f_k$ so that
$$\sum_{I=1}^k \cos(f_i)= \cos\left(\sum_{i=1}^k f_i \right).$$
And what about the infinite case ?

2
votes

0
answers

67
views

### Property of a family of simple polynomials related to the A329369

Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...

9
votes

1
answer

809
views

### Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions

I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...

15
votes

3
answers

1k
views

### Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?

So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...

7
votes

0
answers

218
views

### How are connected the hypergeometric motives of WZ-Pairs?

If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...

4
votes

1
answer

137
views

### Closed form for the A110501 (unsigned Genocchi numbers (of first kind) of even index)

Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here
$$
a(n) = \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2i}a(n-i)(-1)^{i-1}, \\
a(1) = 1
...

3
votes

1
answer

222
views

### Do these polynomials with a complex kind of ‘Vieta jumping’ exist for all $k$?

Inspired by a recent question about sequences defined by $s_{n+1}=s_n^2-s_{n-1}^2$, I started wondering whether non trivial real or complex cycles of any length $k\geqslant3$ fixed by such a sequence ...