# Questions tagged [sequences-and-series]

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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, $a_{i+1}=a_{i}+... • 207 2 votes 0 answers 21 views ### On doubling or addition formulas for the sequence$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)$... • 24.8k 0 votes 1 answer 85 views ### Summation of binomial coefficients with alternating signs For a fixed$\alpha > 1$and integer$n$, I want to provide some bounds or scaling results for the following summations $$S_1(n,\alpha) = \sum_{k = 1}^{n} {n \choose k} (-1)^{k + 1} k / (\alpha k + ... 4 votes 2 answers 281 views +50 ### Conjectured Somos-like closed form of recurrences with polynomial coefficients From Our short paper For polynomial F with integer coefficients, define the recurrence f(n)=F(n,f(n-1),f(n-2),...,f(n-d)). We conjecture that f(n) satisfy Somos like sequence f(n)=\frac{G(f(n-1)... • 24.8k 4 votes 1 answer 143 views ### Closed form expression for \sum_{n=0}^{\infty} J_n^2(x) \cos(ny), where J_n(x) is the Bessel function of order n Anyone can find/calculate a closed form expression for the sum$$ \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}. $$Let a_1(n,p,q) be the family of ... • 4,130 1 vote 1 answer 131 views ### An inequality about binomial distribution Statement Assume that \sigma,R\in (1,+\infty), N\in\mathbb{N}^*, p\in (0,1), n_1\in\{0,1,2,\cdots,N-1\}. Prove or disprove that$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$... • 35 0 votes 1 answer 77 views ### Do disjunctive sequences eventually get palindromic at some point? I have a friend who is very interested in math and has been thinking about a problem involving disjunctive sequences. For his birthday, I would like to give him an answer to his question, either by ... 3 votes 0 answers 224 views ### Exotic series for some mathematical constants from String Theory Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for \pi, Apéry's constant \zeta(3) and ... • 2,720 0 votes 0 answers 128 views ### Integer coefficients and continued fractions Let a(n,p,q) be the family of integer sequences such that ordinary generating functions for it are \frac{1}{G_1(0,x)} where G_1(0,x) are continued fractions such that$$ G_1(j,x)=1-\cfrac{(qj+1)... • 4,130 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$n$... • 509 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 $$t\tan\left(\frac{t}{2}\right)=\sum\limits_{n=1}^{\infty}(-1)^n G_{2n}\... • 4,130 2 votes 1 answer 138 views ### R-recursion for Fibonacci numbers using signed Catalan numbers Let F_n be A000045 (i.e., Fibonacci numbers). Here$$ 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}. $$Let ... • 4,130 1 vote 1 answer 54 views ### Simplest way to generate integer coefficients with row sums equal to the terms of an arbitrary given sequence Let f(n) be an arbitrary function. Let \operatorname{wt}(n) be A000120 (i.e., number of ones in the binary expansion of n). Here$$ \operatorname{wt}(2n+1) = \operatorname{wt}(n) + 1, \\ \... • 4,130 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 ... • 4,130 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 ... • 24.8k 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 $$\ell(n) = \left\lfloor\log_2 n\right\... • 4,130 1 vote 0 answers 84 views ### Closed form for the A357990 using A329369 and generalised A373183 Let$$ \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)... • 4,130 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 ... • 11 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. ... • 509 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 $$A(x) = \sum\limits_{k=0}^{\infty} \frac{a(k)}{... • 4,130 0 votes 0 answers 121 views ### How to prove the convergence of the following series involving Gamma function? Consider the following result（d denotes the dimensions and 0<t<T）$$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 ... • 29 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} $: $$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \... • 509 2 votes 1 answer 165 views ### Simplification of the closed form for the A329369 Let s(n,k) be a (signed) Stirling number of the first kind. Let {n \brace k} be a Stirling number of the second kind. Let$$ f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace ... • 4,130 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 kbe 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}... • 4,130 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 \... • 1,269 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 ... • 6,205 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 ... • 4,130 29 votes 1 answer 3k views ### 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 ... • 1,429 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 ... • 509 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 $$Let \... • 4,130 65 votes 8 answers 7k views ### Possible new series for \pi In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for \pi:$$\pi = 4 + \... • 80.9k 1 vote 0 answers 54 views ### Simple recursion for the A329369 using Stirling numbers of both kinds Lets(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}$with ... • 4,130 0 votes 0 answers 38 views ### Mellin transform of confluent Lauricella hypergeometric function The$F_D^{(n)} $Lauricella's hypergeometric function can be defined as follow $$F_D\left(a,b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(a\right)_{m_1+\cdots+... • 111 1 vote 0 answers 131 views ### Sequence that sums up to A000153 Let a(n) be A329369 (i.e, number of permutations of {1,2,...,m} with excedance set constructed by taking m-i (0 < i < m) if b(i-1) = 1 where b(k)b(k-1)\cdots b(1)b(0) (0 \leqslant k ... • 4,130 3 votes 0 answers 86 views ### Recursion for reversed rows of the A373183 using unsigned Stirling numbers of the first kind Let \left[{n \atop k}\right] be unsigned Stirling numbers of the first kind. Here$$ \left[{n \atop k}\right] = (n-1)\left[{n-1 \atop k}\right] + \left[{n-1 \atop k-1}\right], \\ \left[{n \atop 0}\... • 4,130 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 ... • 509 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 $$T(2n+1, k) = kT(n, k) + T(n, k-1), \\ T(2n, k) = kT(n, k) + T(n, k-1) - \frac{T(2n, k-1) + T(n, k-1)}{k-1}, \\ T(n, 1) = T(0, 2) = 1$$ ... • 4,130 0 votes 0 answers 96 views ### integral of exponential of Fourier series I have encountered the following integral: $$\int_0^{1} e^{-i F(x)} dx, \quad F(x) = \sum_{k=1}^L a_k \sin(2\pi k x) + b_k \cos(2\pi k x).$$ 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 \...
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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 ?