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**22**

votes

**3**answers

2k views

### Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question):
The Riddle:
We assume ...

**7**

votes

**1**answer

1k views

### Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...

**2**

votes

**2**answers

561 views

### Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\...

**104**

votes

**3**answers

11k views

### Convergence of $\sum(n^3\sin^2n)^{-1}$

I saw a while ago in a book by Clifford Pickover, that whether $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open.
I would think that the question of its convergence is really ...

**18**

votes

**1**answer

500 views

### Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...

**9**

votes

**1**answer

948 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**9**

votes

**1**answer

447 views

### Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...

**7**

votes

**3**answers

491 views

### Asymptotic formulas for Monster-related modular functions?

Define the following,
$$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$
$$j_{2A}(\tau) =\Big(\big(...

**11**

votes

**1**answer

1k views

### Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...

**3**

votes

**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 ...

**2**

votes

**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 ...

**78**

votes

**8**answers

6k views

### Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$?
(I think it is.) If so, how would one prove this? (To confirm: This is the power
series for $e^x$, except with the ...

**38**

votes

**2**answers

1k views

### Numbers that are generic w.r.t. exponentiation

This is a follow-up to my old question Number of distinct values taken by $x\hat{\phantom{\hat{}}}x\hat{\phantom{\hat{}}}\dots\hat{\phantom{\hat{}}}x$ with parentheses inserted in all possible ways.
...

**40**

votes

**2**answers

3k views

### Alternating sum of square roots of binomial coefficients

Let
$$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$
It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...

**28**

votes

**12**answers

4k views

### What Are Some Naturally-Occurring High-Degree Polynomials?

To construct J. H. Conway's look-and-say sequence, begin by putting down a 1 as the first entry. The other entries are found by saying the previous entry aloud, and writing what you hear.
...

**21**

votes

**4**answers

2k views

### Does this sequence always give an integer?

It is known that the $k$-Somos sequences always give integers for $2\le k\le 7$.
For example, the $6$-Somos sequence is defined as the following :
$$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot ...

**24**

votes

**4**answers

1k views

### Asymptotic growth of a certain integer sequence

Some time ago, while putting my nose in the Sloane's Online Encyclopedia of Integer Sequences, I came over the sequence A019568 defined as follows:
$a(n):=$ the smallest positive integer $k$ such
...

**37**

votes

**4**answers

2k views

### Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula.
I. Given the fundamental unit $\...

**20**

votes

**2**answers

2k views

### Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?

Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and
$$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$
See ...

**17**

votes

**1**answer

1k views

### Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$.
1. Define the following sequences,
$$\begin{aligned}
u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\
u_2(k) &...

**27**

votes

**5**answers

1k views

### How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This will not be altogether unrelated to this earlier question.
For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...

**20**

votes

**4**answers

1k views

### Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...

**20**

votes

**9**answers

6k views

### What is the indefinite sum of tan(x)?

What is the indefinite sum of the tangent function, that is, the function $T$ for which
$\Delta_x T = T(x + 1) - T(x) = \tan(x)$
Of course, there are infinitely many answers, who all differ by a ...

**13**

votes

**2**answers

918 views

### a weird sequence with a non-integral term

Define a sequence $(a_n)_{n \geq 1}$ by $$na_n = 2 + \sum_{i = 1}^{n - 1} a_i^2.$$
(In particular, $a_1 = 2$.)
How can you show - preferably without using a pc! - that not all terms of the sequence ...

**9**

votes

**2**answers

2k views

### Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:
I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...

**9**

votes

**2**answers

452 views

### Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum:
$$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$
...

**10**

votes

**1**answer

916 views

### Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made:
Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened.
Define ...

**7**

votes

**2**answers

577 views

### Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function:
$$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$
Newton series expansion of a function:
$$f(x)...

**5**

votes

**1**answer

530 views

### Is this known alternating sum for Euler's constant?

This probably is known, but Wolfram Alpha doesn't recognize it
and couldn't find it in Mathworld (there is something close,
but using floor).
We have
$\lim_{s \to 1} (\zeta(s)-1/(s-1)) = \gamma$
...

**5**

votes

**2**answers

946 views

### Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
I asked ...

**3**

votes

**1**answer

223 views

### A question on graphic sequences

Let $G$ be a graph and $d_{G}(u)$ denotes degree of a vertex $u$ in $G$. Consider the next multiset $$M_{G}:=\{|d_{G}(u)-d_{G}(v)|:\ uv\in E(G)\}.$$
Conjecture: $M_{G}$ is graphical for every $G$.
...

**11**

votes

**1**answer

286 views

### Cantor set intersecting a geometric sequence

I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...

**9**

votes

**4**answers

2k views

### Unique limits of sequences plus what implies Hausdorff?

It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff.
What I am ...

**8**

votes

**1**answer

326 views

### The sum of a series

Let $0< \alpha <1$ and $q>1.$
Consider the (alternating) series: $$
\sum_{k=1}^\infty
(-1)^k \frac{q^k (q^k-1)^\alpha}{(q^k-1)\dots (q-1)}.$$
Denote its sum by $f(q,\alpha).$
Prove (or ...

**3**

votes

**2**answers

405 views

### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

**1**

vote

**1**answer

338 views

### Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...

**9**

votes

**1**answer

267 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 ...

**8**

votes

**1**answer

304 views

### Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below:
$$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$
$$\frac{b_k^2}{b_{k-1}b_{k+1}}=1+\frac{\...

**6**

votes

**3**answers

522 views

### Is there a closed formula for the generating function of some trinomial coefficients?

We learn in calculus how to obtain a sum of binomial coefficients $\frac{(2d)!}{(d!)^2}$ in terms of a generating function
$\sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d$
by the Taylor series of $(1-4x)^...

**5**

votes

**2**answers

349 views

### How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$?

I have to estimate the expression $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for $\kappa$ very small $\kappa \sim 10^{-19}$ and $N$ very large $N\sim 10^{26}$ and $a$ arbitrary $a=1, \ldots, N$. I do not ...

**4**

votes

**4**answers

523 views

### What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...

**6**

votes

**2**answers

290 views

### Determining when combinatorial sums are zero

To keep things simple with a specific example, we ask:
Prove that $\displaystyle\ a_n:=\frac{1}{n!}\sum_{k=0}^n \binom{n}{k} \frac{1}{k!} (-1)^{n-k}$ is zero if and only if $n=1$. (Or find a counter-...

**5**

votes

**1**answer

443 views

### Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

I am not familiar with newforms, so this may not make any sense.
OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3]
Numerical evidence suggest that up to $10^5$
$$ \...

**4**

votes

**1**answer

638 views

### Extension of the Jacobi triple product identity

The Jacobi triple product and the mathematical identity of it is:
$$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$
I would ...

**3**

votes

**2**answers

395 views

### Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181:
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...
Primes $p$ ...

**2**

votes

**1**answer

150 views

### a second order difference equation related to a real polynomials which seems to have only real roots

I am seeking solutions to the following difference equation:
$$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$
where $A>B>0$.
This equation is related to a real polynomial (see here) which I want to ...

**2**

votes

**0**answers

109 views

### Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...

**1**

vote

**0**answers

64 views

### Algebraic Relations between $G(q)$ and $H(q)$

I had posted this originally on MSE, but got no response at all hence posting the same here.
In his paper "Algebraic Relations between Certain Infinite Products" (Proceedings of the London ...

**0**

votes

**1**answer

123 views

### Energy of repeated filter

For given sequences $a=(a_1, a_2, \cdots)$ and $b=(b_1, b_2, \cdots)$, define
$$a \star b$$ as the convolution. Formally, $$c=a \star b$$ implies the $i$th element of $c$, $c_i$, satisfies the ...

**0**

votes

**0**answers

76 views

### Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem:
Is there a method to extend the notion ...