The sequences-and-series tag has no usage guidance.

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**1**answer

64 views

### GCD for two Cullen numbers

The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?

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**0**answers

92 views

### Summing a series of integrals [on hold]

EDIT: This IS related to my research (investigating representations of the harmonic mean)and I gave the wrong formula for the sum the first time around. The sum formula has been amended below.
I ...

**29**

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**2**answers

648 views

### $x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?

This question was asked on MathStackexchange here, but there was no answer, so I am asking it here.
Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...

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**2**answers

173 views

### $\sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} =? $

Can anyone sum up this series?
$f(z, t) = \sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} . $
In the mathematical sense, each term of this series is of modulus 1, and the series is not convergent. ...

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**0**answers

61 views

### 2D sequence of integers [closed]

I found the following sequence of integers $a_n^k$, where $n$ and $k$ to extend to infinity. Each row are coefficients of a polynomial, similar to the coefficients of the Legendre polynomials. The ...

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**2**answers

208 views

### Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...

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88 views

### Factorial Series

Is there a closed form expression for
$$ \sum_{k=n}^\infty\frac{k!^2}{(k+x)(k-n)!(k+n+1)!} $$
where $0<x<1$ ?
(For $n=0$, I know that
$$\sum_{k=0}^\infty\frac{1}{(k+x)(k+1)}=\frac{\psi(x)+\...

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**1**answer

136 views

### Ellipsoidal harmonics - A Series expansion for Lame functions of the second kind

$\underline{Intro \;to \;skip}$
In the theory of ellipsoidal harmonics, Lame functions of the second kind $F_n$ arise as the second linearly independent solution (the first being Lame functions of ...

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144 views

### Can someone explain this appearance of the Fibonacci series in the formula of the Fibonacci series? [closed]

I have found the Fibonacci series as a function. The function is as follows :-
$$F(x) = 1 - 0×f_1(x) + 1×f_2(x) - 1×f_3(x) + 2×f_4(x) - 3×f_5(x) + 5×f_6(x) - 8×f_7(x) + 13×f_8(x) - 21×f_9(x) + 34×f_{...

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**0**answers

157 views

### Analytic expression for the Tsirelson bound of the I3322 inequality?

Finding Tsirelson bounds for Bell inequalities is a well-loved problem in quantum information theory. A famous case where it is still open is for the I3322 inequality. In this paper Pál and Vértesi ...

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**0**answers

374 views

### Number Theory and d-Self-Contained Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...

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**1**answer

153 views

### Solving for f given constraint involving f(x, y) and f(xy, y)

I am interested in a weighted version of the Catalan numbers. The generating function for this case,
$$ f(x, y) = \sum_s \sum_n f_{s n} x^s y^n $$
(where the $y^n$ term is the weight), obeys the ...

**2**

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**1**answer

2k views

### Explicit formula for Euler zigzag numbers(Up/down numbers)

I have derived an explicit formula for the Euler zigzag numbers, the number of alternating permutations for n elements:
$$A_n = i^{n+1}\sum _{k=1}^{n+1} \sum _{j=0}^k {k\choose{j}} \frac{(-1)^j(k-2j)^...

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183 views

### Number Theory and p-Power-Partitioned Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, we're going to define its digits-partition as the next set $D_{N} = \bigcup_{j=1}^{n}\bigcup_{k=1}^{p(a_{j})}\{(P_{k},j)\}$, where each pair $(...

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**1**answer

132 views

### Methods to tackle this series and get to the limit?

Take a look at the averaging sum
$$\frac{\pi}{n}\sum_{k=1}^n\;\exp{(-\sin\theta_k)}\cdot \sin(\theta_k +\cos\theta_k)\, \qquad\text{where }\;\theta_k=(2k-1)\frac{\pi}{2n}$$
depending on $n\in\...

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**0**answers

46 views

### The connection between the length of Fibonacci $p$-step numbers and it's limit values

One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows
\begin{equation}\label{cp26}
F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...

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**2**answers

313 views

### Needing proof of convergence for a sequence

Let $\left\{u_i\right\}_{i=1}^\infty$ be a sequence of real vectors (i.e. $u_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. ...

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**1**answer

216 views

### bounding derivative of a sequence

I've been banging my head against the wall on this one ... define a sequence of polynomials $q_n$ by $q_0 = 0$ and $$q_{n+1} = q_n + .5(t^2 - q_n^2).$$ If $q_n \leq t$ on $[0,1]$ then $$.5(t^2 - q_n^2)...

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**1**answer

387 views

### Does 53 diverge to infinity in this Collatz-like sequence?

This function has been explored a bit at MSE (over the past week):
\begin{eqnarray}
f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\
f(n) &=& \lfloor n/4 \rfloor \; \textrm{...

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**1**answer

288 views

### Summation of series involving $\sinh$ of a square root

Consider the following series:
$$
S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})}
$$
From the physical ...

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**2**answers

258 views

### Finding a closed form for $\sum_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3}$

I'm looking for a closed form for the expression
$$
\sum_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3}
$$
I know that Ramanujan gave the following closed form for a similar expression
$$
\sum_{k=1}^{\infty}\...

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**1**answer

752 views

### sum of the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$?

Hi,
We know that the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$ is convergent and it is oscillating. and numerically it is almost 0.6048986434.
I want to know what is the exact limit ...

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284 views

### Rewriting a series $\sum_{n=0}^\infty \frac{1}{n!}(\Delta^\varepsilon)^n a_n$ in the form $\sum_{n=0}^\infty c_n \varepsilon^n$

I would like to rewrite the series $$\sum_{n=0}^\infty \frac{1}{n!}(\Delta^\varepsilon)^n a_n,$$ where $\Delta^\varepsilon=\sum_{k=1}^\infty \varepsilon^k b_k$, as a series in $\varepsilon$ $$\sum_{n=...

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**1**answer

625 views

### How to find the coefficients of a poor-converging series?

I have the series
$\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$
and the boundary conditions
$\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(...

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**0**answers

67 views

### How to prove that the convergence of $\sum_{n=1}^{\infty} \frac{\sec^a n}{n^c}$ implies that of $\sum_{n=1}^{\infty} \frac{\csc^a n}{n^c}$

The most general thing I've gotten is that the absolute convergence of $$\sum_{n=1}^{\infty} \frac{\csc^a (n + x)}{n^c}$$ implies that of $$\sum_{n=1}^{\infty} \frac{\csc^a \left(\frac{m}{2} n + \...

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50 views

### a two dimensional integer bijection

I want to try to find a specific two-dimensional linear integer bijection. This is to be used in a double sum rearrangement. It's kind of complicated, but I would really appreciate if anybody has any ...

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**0**answers

48 views

### Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694
Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...

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**1**answer

250 views

### Is this function $\mathbb{Q}$ periodic?

Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ?
$$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; \sum\...

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**1**answer

177 views

### On construction of a $\mathbb{Q}$ periodic function with Fourier series

Taking $f$ a function decreasing exponentially at infinity we can consider the periodic function given by following Fourier series:
$$F(x)= \sum\limits_{n =1}^{\infty} f(n) e^{2 i \pi n x}$$
Using ...

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**0**answers

183 views

### First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...

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**0**answers

74 views

### Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...

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**2**answers

486 views

### How many integers are of the form $n/d(n)$, where $d(n)$ is the number of divisors of $n$?

This was post by me on Maths SE: but it did not get any solution.
Some months ago I made the following conjecture -
Let $d(n)$ denote the number of divisors of $n $.
Then let $N$ be a number such ...

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**0**answers

48 views

### Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations.
Let $A$ be an $...

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**1**answer

134 views

### Generalized Equal Distribution Kolakoski Sequence Conjecture

If we let $\operatorname{Kol}(a_1,\dots,a_n)$ be the run sequence determined by the rules of Kolakoski Frequencies, we ask is there a sequence of $\operatorname{Kol}$ that DOES NOT obey the $1/n$ ...

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**1**answer

167 views

### For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...

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**1**answer

63 views

### Inequality implies locally uniform convergence of a series

We have the inequality
$$\alpha_n(t) \le 4\pi(3\pi)^{1/3} \exp\left\{\int_0^t(1+3F_P(\sigma)) \, d\sigma \right\} \cdot \int_0^t P(\sigma)^2(3CD_1^2)^{1/3}\alpha_{n-1}(\sigma) \, d\sigma$$
for $n=2,3,\...

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240 views

### How to estimate a specific infinite matrix sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...

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78 views

### How to estimate $\prod_{t=1}^{N}\frac{1}{2-z^t}$ for large $N$?

Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$?
Can anyone find an approximate closed form for
$$
\frac{\mathrm{d}^k}{\mathrm{d}z^k}\prod_{t=1}^{N}...

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296 views

### Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$? [closed]

What is the closed form of
$$\sum_{k=0}^n \frac{x^k}{k!}$$
as a function of $x$ and $n$?
Knowing that it converges to $e^x$ when $n\to \infty$.

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235 views

### Moment problem on [-1,1]: necessary and sufficient conditions

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that
$$
s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?
$$
...

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**1**answer

148 views

### Does $\sum_n \frac{\sin n}n$ converge absolutely? [closed]

Using Dirichlet's test, one can prove that $\sum_{n\geq 1} \frac{\sin n}n$ converges. Does it converge absolutely?

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**2**answers

2k views

### Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...

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**4**answers

427 views

### The coefficient of a specific monomial of the following polynomial

Let the real polynomial
$$f_{a,b,c}(x_1,x_2,x_3)=(x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1},$$
where $a,b,c$ are nonnegative integers.
Let $m_{a,b,c}$ be the coefficient of the monomial $x_1^{...

**2**

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**1**answer

101 views

### Sufficient conditions for weak majorization

Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive integer $p$,
$$
\sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p.
$$
Question: Is ...

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**1**answer

266 views

### How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently

Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute
$$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$
One option is to simply ...

**4**

votes

**1**answer

186 views

### Non-equivalence of admitting different types of bases in Banach spaces

Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...

**1**

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**2**answers

158 views

### aproximate sum involving binomial coefficients

I have the problem for computing the j-derivative of a logarithm, with $j\gg1$
\begin{equation}
c_j=\left.\frac{\partial^j}{\partial s^j}\log\left(1+Ae^s+Be^{2s}\right)\right|_{s=0},
\end{equation}
...

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**5**answers

631 views

### Asymptotics of a recurrence relation

The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation:
$$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$
where, $[x]$ is the nearest integer to $x$ not exceeding ...

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votes

**0**answers

103 views

### Enumerating the number of degree d curves tangent to a planar conic

This question is based on a special case of the Coparaso Harris formula, as described in Counting curves on rational surfaces - R. Vakil.
Let $E$ be a non-singular planar conic.
Then every degree $d$...

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**2**answers

849 views

### Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$:
\begin{align}\frac{1}{2\pi\sqrt{2}} &= \frac{1103}{99^{2}} +
\frac{...