Questions tagged [sequences-and-series]

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On partitions into distinct parts and binary

Let $a(n)$ be A000009 (i.e., number of partitions of $n$ into distinct parts or number of partitions of $n$ into odd parts). Let $$ b(n) = \sum\limits_{i=0}^{n} a(i) $$ Let $$ \ell(n) = \left\lfloor\...
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Bound on a two-dimensional recursive series

For $n,k\in\mathbb{N}$, let $f(n,k)$ be defined as follows. If $n \geq k$ and $n > 2$, then $$ f(n,k) = \frac{k(n-k)}{n(n-1)}f(n-2,k-1) + \frac{k(k-1)}{n(n-1)}f(n-2,k-2) + \frac{n-k}{n}f(n-1,k) + \...
macat's user avatar
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A sequence linked to irrationality

Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by : $$u_0 = x$$ $$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
Azoth's user avatar
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Given $F[N,M]=\sum_{m=0}^{N-1}(-1)^{N-1-m}(m+1)^M)/(m!(N-1-m)!)$, show $F[N,N-1]=1$ and $F[N,M]=0$ for $M<N-1$

The function defined by $$ F[N,M]=\sum_{m=0}^{N-1}\frac{(-1)^{N-1-m}(m+1)^M}{m!(N-1-m)!} $$ where $N,M$ are positive integers. I want to show $$ F[N,N-1]=1,\ F[N,M]=0 $$ for $N>2$ and $M<N-1$. ...
Guoqing's user avatar
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3 votes
1 answer
208 views

How to find the coefficient of $x^k$ in the expression $\prod_{p=2}^n (1+xp) $

I got this general formula for $ n\in N$ (I showed it here) $$\int_0^1 \left(\frac{x}{1-x} \ln x \right)^n dx=n \sum_{p=0}^{n-1}a(n,p+1) (-1)^{n-p} \zeta(p+2)+n! $$ where $a(n,k)$ is the coefficient ...
Faoler's user avatar
  • 431
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0 answers
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Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$

I posted this question on Math Stack Exchange, but there were no helpful comments or answers https://math.stackexchange.com/q/4874446/1298448 How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
Martin.s's user avatar
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joint rank sequences

An algebraic question I have been working on led me to a sequence that appears in OEIS as A186355: "adjusted joint rank sequence of $(f(i))$ and $(g(j))$ with $f(i)$ before $g(j)$ when $f(i)=g(j)$...
Vladimir Dotsenko's user avatar
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55 views

Pairs of permutations such that $p(n)<2^k$ iff $n<2^k$

Let $p(n)$ be an arbitrary permutation of natural numbers such that $p(n)<2^k$ iff $n<2^k$. Let $q(n)$ be an inverse permutation of $p(n)$. Let $$ \ell(n)=\left\lfloor\log_2 n\right\rfloor $$ ...
Notamathematician's user avatar
4 votes
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69 views

Formula and smallest solution for the A260711

Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$). The sequence begins with $$ 8, 16, 32, 48, 64, ...
Notamathematician's user avatar
3 votes
0 answers
190 views

A problem about the series $\sin(n^p)$ [closed]

Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$ is divergent
adobereader's user avatar
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On a numbers $k$ with specific $2$-adic valuation

Let $a(n)$ be A002326 (i.e., multiplicative order of $2 \operatorname{mod} 2n+1$). Let $b(n)$ be A179382 (i.e., the smallest period of pseudo-arithmetic progression with initial term $1$ and ...
Notamathematician's user avatar
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Formula for individual term of the Proth numbers

Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$. The sequence begins with $$ 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129 $$...
Notamathematician's user avatar
2 votes
0 answers
67 views

Possible subsequence of the A110978

Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
Notamathematician's user avatar
4 votes
2 answers
287 views

Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$

I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
Sayan Dutta's user avatar
3 votes
0 answers
238 views

On thickness of binary polynomials

OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
Sayan Dutta's user avatar
3 votes
1 answer
200 views

Partition numbers and Gaussian binomial coefficient

Let $a(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers). Let $T(n, k)$ be A083906. Here $$ T(n, k) = [q^k]\sum\limits_{m=0}^{n} \binom{n}{m}_q $$ where $\binom{n}{m}_q$ ...
Notamathematician's user avatar
1 vote
0 answers
97 views

Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions

For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$. For $D \le 6$, sage finds closed form in terms of hypergeometric functions at algrebraic arguments and fails to find closed ...
joro's user avatar
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On a Fibonacci and binary

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 $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor $$ Let $$ T(n, k) = \left\lfloor\frac{n}{2^k}\...
Notamathematician's user avatar
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61 views

Closed form of the sum of a fast-converging series

The chi-square distribution with $k$ degrees of freedom is $$ f(x)\, dx = \frac1{\Gamma(k/2)} \left( \frac x2\right)^{(k/2)-1} e^{-x/2} \left( \frac{dx} 2 \right) \qquad \text{for } x>0. $$ This ...
Michael Hardy's user avatar
2 votes
0 answers
80 views

Splitting natural numbers into subsets with sums equal to A066258

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 $a(n)$ be A066258 i.e. $$ a(n) = F(n)^2F(n+1) $$ Let $b(n)$ be A345253 i.e. maximal ...
Notamathematician's user avatar
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$R$-recursion for the A007165

Let $a(n)$ be A007165 i.e. number of $P$-graphs with $2n$ edges. Here ordinary generating function $A(x)$ satisfies $$ A(x) = \frac{(1 + xA(x))(1 + 2xA(x))}{1 + 2xA(x) - (xA(x))^2} $$ Let $$ R(n, q) = ...
Notamathematician's user avatar
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1 answer
108 views

Limit of $F_{n}(\lfloor{nx}\rfloor)$ where $ F_{n}(k)=G_{n}(k)+H_{n}(k)F_{n}(k+1) $ and $F_{n}(n)=\mu.$

The following conjecture is inspired by asymptotic results in generalizations of the secretary problem. CONJECTURE Consider a sequence of functions {$F_n$} with $F_{n}:[0,n]\cap \mathbb{Z}\rightarrow\...
José María Grau Ribas's user avatar
32 votes
3 answers
2k views

Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

Updated on Feb.16.2024 Fortunately for three of these series, Eqs. (1), (3) and (4), I have found a proof using classical methods which I placed in the Answers section below. (I doubt that there is a ...
Jorge Zuniga's user avatar
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1 vote
0 answers
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$R$-recursion for the A036765

Let $a(n)$ be A036765 i.e. number of ordered rooted trees with $n$ non-root nodes and all outdegrees $\leqslant 3$. Here $$ a(n) = \frac{1}{n+1}\sum\limits_{j=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\...
Notamathematician's user avatar
2 votes
1 answer
139 views

$R$-recursion for the A143017

Let $a(n)$ be A143017 i.e. number of $\{2-1-3, 2'^e-31\}$-avoiding permutations of size $n$ (see definition in the Elizalde paper). Here $$ a(n) = \frac{1}{n}\sum\limits_{k=0}^{\left\lfloor\frac{n}{...
Notamathematician's user avatar
5 votes
1 answer
235 views

How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?

I tried to find an inverse Laplace transform by series as follows $$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$ and by ...
Faoler's user avatar
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1 vote
1 answer
129 views

Strongly regular binary sequences

Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote the set of non-negative integers. If $n\in\mathbb{N}$ we let $[n] = \{0,\ldots,n-1\}$. For $A \subseteq \mathbb{N}$ we let $$\mu^+(A) = \lim\sup_{n\to\infty}\...
Dominic van der Zypen's user avatar
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0 answers
150 views

Triangular and pentagonal numbers in $q$-series

Consider the following two infinite series $$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\, \sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
T. Amdeberhan's user avatar
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0 answers
154 views

Relationship between Fourier inversion theorem and convergence of "nested" Fourier series representations of $f(x)$

$\DeclareMathOperator\erf{erf}\DeclareMathOperator\sech{sech}\DeclareMathOperator\sgn{sgn}\DeclareMathOperator\sinc{sinc}$This is a cross-post of a question I posted on MSE a couple of weeks ago which ...
Steven Clark's user avatar
  • 1,061
1 vote
1 answer
89 views

General case of the some $R$-recursions

Let $f(n)$ be an arbitrary function. Let $a(n)$ be an integer sequence such that its ordinary generating function satisfies $$ A(x)=\sum\limits_{i=0}^{\infty}\frac{x^i}{\prod\limits_{j=0}^{i}(1-f(j)x)...
Notamathematician's user avatar
15 votes
1 answer
1k views

The Mompox Sequence: are all its terms different?

The Mompox Sequence, $a(n):=1, 2, 6, 24, 120, 20, \ldots$ (OEIS A008336), is the sequence of positive integers whose first term is $1$, and in which the $n$-th term (after the first one) equals the ...
Bernardo Recamán Santos's user avatar
6 votes
2 answers
1k views

5n+1 sequence starting at 7

Consider the following variant of the Collatz function: $f:\mathbb N\rightarrow\mathbb N$ is defined by \begin{equation} f(n):=\begin{cases} n/2 & \text{if $n$ is even}\\ 5n+1 & \...
Riemann's user avatar
  • 537
10 votes
2 answers
995 views

Does a conditionally convergent sum with random signs converge almost surely?

Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
Nate River's user avatar
  • 4,802
8 votes
1 answer
1k views

When can a sum be re-signed to converge to any limit?

Let $a_n$ be a sequence of positive real numbers with $\sum a_n < \infty$. What are the necessary and sufficient conditions for the following to hold? For any $S \in \mathbb R$ with $-\sum a_n \...
Nate River's user avatar
  • 4,802
2 votes
1 answer
145 views

Growth rate of elementary sequences

We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
António Borges Santos's user avatar
4 votes
1 answer
274 views

3 divides coefficents of this $q$-series

Denote $\phi(q):=\prod_{j\geq1}(1-q^j)$ and let $\xi=e^{\frac{2\pi i}3}$ be a cube root of unity. Define the sequence $u(n)$ by $$\prod_{n\geq1}\prod_{s=1}^2(1-q^n\xi^{ns})(1-q^{2n}\xi^{ns}) =\sum_{n\...
T. Amdeberhan's user avatar
2 votes
5 answers
904 views

Binomial series

I am interested in the limit $\frac{\sum_{k=0}^n \sqrt{k}\cdot\binom{n}{k}}{\sqrt{n}\cdot2^n}$ as $n$ goes to infinity. Any reference or argument? In general what do we know about the asymptotic ...
Morteza's user avatar
  • 628
3 votes
1 answer
362 views

Generalized harmonic numbers and Riemann zeta function

The $n$-th harmonic number is defined as $$ H_{n}=\sum\limits_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(m)}=\sum\limits_{k=1}^{n}\frac{1}{k^m}. $$ It is ...
Notamathematician's user avatar
3 votes
1 answer
284 views

Smallest number of subsets whose squares cover the whole square

Let $2 \leq k \leq n$ be integers, let $[n] := \{1,2,\ldots,n\}$, and for a subset $A \subseteq [n]$ let $A^2 := A \times A$ be the Cartesian product of $A$ with itself and let $|A|$ denote the ...
Nathaniel Johnston's user avatar
4 votes
0 answers
163 views

Growth rate of a recursively defined sequence

For a side project with a friend (having to do with fractional iterates of the exponential function), we're looking at a sequence $a_n$ defined recursively by equations $a_1 = 1$ and $$a_n = \sum_{k=1}...
Todd Trimble's user avatar
  • 52.3k
1 vote
1 answer
123 views

Is a bounded convex function $g$ that is non-negative on this particular convex set also non-negative on the its closure?

Setup : Let $S$ be the simplex on $\mathbb N$, i.e. the set of probability distribution on the natural numbers. Suppose we have $p\in S$ such that, for all $n\geq 1$, $p_n > 0$. For any $\emptyset\...
P. Quinton's user avatar
1 vote
1 answer
84 views

$R$-recursion for the A307389

Let $a(n)$ be A307389 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies $$ A(x)=\exp\left(\frac{\exp(2x)-2\exp(x)+2x+1}{2}\right) $$ The sequence begins with $$ 1,...
Notamathematician's user avatar
43 votes
3 answers
2k views

Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$?

For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that $$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$ Let \begin{...
math110's user avatar
  • 4,220
1 vote
1 answer
108 views

Properties of the relatively bounded probability distributions on the simplex over the natural numbers

Setup : Let $S$ be the simplex over $\mathbb N$, i.e. the set of probability distribution over $\mathbb N$. Let $p\in S$ be such that $0 < p_n$ for all $n\in \mathbb N$. Let $H$ be the Hilbert ...
P. Quinton's user avatar
1 vote
0 answers
71 views

Slightly modified program for the A345253 such that specific partial sums equal A066258

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 $a(n)$ be A345253 i.e. maximal Fibonacci tree: arrangement of the positive integers as ...
Notamathematician's user avatar
6 votes
1 answer
303 views

A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Could this be proven?

This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function: $$\xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\...
Agno's user avatar
  • 4,179
4 votes
1 answer
182 views

Partition numbers as the specific sums of the A161511

Let $p(n)$ be A000041 i.e. number of partitions of $n$ (the partition numbers). Let $$ \ell(n)=\left\lfloor\log_2 n\right\rfloor $$ Let $a(n)$ be A161511 i.e. number of $1\cdots0$ pairs in the ...
Notamathematician's user avatar
1 vote
0 answers
160 views

Amateur Exploring the 'Honeycomb Sequence': A Novel Mathematical Pattern Derived from Pascal's Triangle [closed]

I am an amateur, and for fun, I was studying a specific number sequence I called the "Honeycomb Sequence," derived from hexagonal patterns in Pascal's Triangle. The sequence involves ...
thomasfreund's user avatar
0 votes
1 answer
96 views

Finding the smallest possible value of $|S_n|$ for Sequence and Series in real analysis [closed]

Let $𝑋_1,𝑋_2, … , 𝑋_𝑛$ be a sequence of independent random variables with $𝑃(𝑋_𝑛 = 4^𝑛) = 𝑃(𝑋_𝑛 = −4^𝑛) = \frac12$. Let $𝑆_𝑛 = 𝑋_1 + 𝑋_2 + ⋯ + 𝑋_𝑛$. If $A_n=\sup\, \{𝑟 ∈ \Bbb R: 𝑃(|...
john22445's user avatar
2 votes
2 answers
246 views

If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?

Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
YC Su's user avatar
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