All Questions
Tagged with nt.number-theory sequences-and-series
612 questions
37
votes
3
answers
2k
views
How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?
For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have
$$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$
Note that this ...
- 215
0
votes
1
answer
129
views
Closed form for $\sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]$ where $\operatorname{wt}(n)$ is the binary weight of $n$
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
Let $a(n,m)$ be the family of integer sequences such that
$$
a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt}(...
- 7,226
8
votes
1
answer
671
views
Infinite series and sum of two squares
Consider the following infinite sequence $a(n)$ generated by
$$\sum_{n\geq0} a(n)q^n
=\frac{\sum_{k\geq0}F(2k+1)q^{\binom{k+1}2}}{\sum_{k\geq0} q^{\binom{k+1}2}}$$
where the $F(2k+1)$ are the odd ...
- 79.9k
10
votes
0
answers
350
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 24.2k
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
0
votes
1
answer
169
views
Partial sums of binomial coefficients and related family of polynomials
Let $a(n)$ be A302117. Here
$$
a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\
a(0) = 0.
$$
Let
$$
T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}.
$$
Let $P_n(z)$ be the family of ...
- 34.3k
1
vote
1
answer
203
views
Structural differences between closed forms of two related infinite products?
In this question, I went quite a bit over the top, so I now tried to rephrase it in a much simpler way.
Take $a \in \mathbb{R}, s \in \mathbb{C}$ and:
$$\displaystyle C(s,a) := \prod_{n=1}^\infty \...
CommunityBot
- 1
6
votes
1
answer
494
views
(Explicit) Tauberian theorems: removing $(\log x/n)$
Say that $\{a_n\}_{n\geq 1}$, $|a_n|\leq 1$, are such that $$\left|\sum_{n\leq x} a_n \log \frac{x}{n}\right|\leq \epsilon x\quad\text{for all $x\geq x_0$.}$$ What sort of bound can we deduce on $S(x)=...
- 20.2k
1
vote
0
answers
57
views
Step back step forward algorithm for A108442
Let $a(n)$ be A108442. Here generating function is $\frac{1}{1-zA(z)}$ where
$$
A(z) = 1 + z(A(z))^2 + z(A(z))^3.
$$
Also
$$
a(n) = \sum\limits_{k=1}^{n}\frac{k}{2n-k}\sum\limits_{i=0}^{n-k} \binom{2n-...
- 4,938
4
votes
1
answer
213
views
Asymptotic behavior of weighted sums involving the fractional part function
Currently, I am studying the asymptotic behavior of sums of the form
\begin{equation}\label{eq1}\tag{1}
\sum_{k=1}^{n-1} f(n-k) \left\{ \frac{n}{k} \right\}
\end{equation}
In this context, based on ...
- 611
2
votes
0
answers
182
views
Algorithm for $\frac{1}{1-x} = \sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n}\frac{1-kx}{1+kx}$
Let $a(n)$ be A208832. Here
$$
\frac{1}{1-x} = \sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n}\frac{1-kx}{1+kx}.
$$
Start with vector $\nu$ of fixed length $m$ with elements $\nu_i = 1$ (that ...
- 175
1
vote
0
answers
161
views
Efficient algorithm for A217061
Let $a(n)$ be A217061. Here
$$
a(n) = \sum\limits_{m=1}^{n}\frac{1}{(m-1)!}\sum\limits_{k=0}^{n-m}(n+k-1)!\sum\limits_{j=0}^{k}\frac{1}{(k-j)!}\sum\limits_{\ell=0}^{j}\frac{2^{\ell-j}(-1)^{\ell+j}s(n-...
- 4,938
9
votes
5
answers
2k
views
Laplace's summation formula
I recently came across the following formula, which is apparently known as Laplace's summation formula:
$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{...
- 12.3k
2
votes
0
answers
67
views
$R$-recursion for A006351
Let $a(n)$ be A006351 (i.e., number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon). Here exponential generating function is $A(x)$ such that $B(x) = ...
- 175
2
votes
0
answers
59
views
$R$-recursion for A338193
Let $a(n)$ be A338193. Here generating function is $A(x)$ such that
$$
A(x) = 1 + \int\frac{\left(\frac{x}{A(x)}\right)'}{\left(\frac{x}{(A(x))^2}\right)'} \, dx.
$$
Let
$$
R(n, q) = \begin{cases}
1 &...
- 4,938
16
votes
0
answers
351
views
The convergence domain of the function $\sum \{n!x\}$
This is a problem from a mathematics competition: Does there exist an irrational number $x$ such that the series
$$\sum_{n=1}^{\infty}\{n!x\}<+\infty$$
where $\{ \}$ means the fractional part of a ...
- 2,858
1
vote
1
answer
92
views
Equivalence of sequences related to A033264
Let $a(n)$ be A033264 (i.e., number of blocks of $\{1,0\}$ in the binary expansion of $n$). Here
$$
a(4n) = a(4n+1) = a(2n), \\
a(4n+2) = a(n)+1, \\
a(4n+3) = a(n), \\
a(0) = 0.
$$
Let
$$
\ell(n) = \...
- 60.5k
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 105k
1
vote
0
answers
89
views
Test for odd prime triples in a $2p-1$ progression
Let $a(n)$ be A057326 (i.e., first member of a prime triple in a $2p-1$ progression).
Let $b(n) = B$ after $n-1$ iterations where we start with $A=n, B=1$ and for $i$ from $1$ to $n-1$ simultaneously ...
- 4,938
2
votes
0
answers
113
views
Numbers of positive terms in polynomials equal A069999
Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known.
Let $P(n,k)$ be ...
- 4,938
5
votes
0
answers
285
views
How do you go about making ranges (for integer variables) independent?
Basic question: say you have a sum
$$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$
where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
- 20.2k
1
vote
1
answer
178
views
Algorithm for A127782
Let $a(n)$ be A127782 (i.e., an integer sequence with generating function $A(x)$ such that $A(x)=1+xA(x+x^2)$). Here
$$
a(n) = \sum\limits_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{...
- 34.3k
1
vote
0
answers
82
views
Generating functions related to generating function of Catalan numbers
Let $C_n$ be A000108 (i.e., Catalan numbers). Here generating function is $C(x)$ such that
$$
C(x) = \frac{1-\sqrt{1-4x}}{2x}.
$$
Let $a(n)$ be an integer sequence with generating function $A(x)$ such ...
- 4,938
1
vote
0
answers
84
views
Coarse well-distributedness/equidistribution of Pell sequence prefixes
I am interested in the distributedness or "mixing" behavior of certain
linear recurrences modulo powers of $2$.
In particular, consider the Pell sequence (https://oeis.org/A000129),
modulo $...
- 11
0
votes
0
answers
65
views
Algorithm and equivalent recursion for A258173 (related to Dyck paths)
Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$.
A Dyck path of semilength $n$ is a $(x,y)$-...
- 4,938
0
votes
1
answer
141
views
Property of composite numbers
Let $a(n)$ be the sequence of composite numbers (starting from $4$). Let
$$b(n)=a(n-1)a(n-2) \operatorname{mod} a(n)$$
Obviously, $b(1)=b(2)=0$.
I conjecture that with the only exception for the $b(3)=...
9
votes
0
answers
258
views
On a continued fraction and vector $\nu$ of length $n$
Please note that this question has been completely reworked in order not to overload it with unnecessary and useless information.
Let $f(n)$ be an arbitrary function with integer values.
Let $a(n)$ ...
- 4,938
2
votes
0
answers
121
views
Solving a system of differential-like equations for reverse Euler-Maclaurin summation
Aim
A particular instance of a rational zeries that has as of yet not been evaluated is:
\begin{align}
Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1}
\end{align}
This sum ...
- 4,829
2
votes
0
answers
61
views
Algorithm for main diagonal of integer coefficients associated with Schroeder numbers
Let $T_q(n, k)$ be an integer table such that
$$T_q(n, k) = \begin{cases}
1 & \textrm{if } n = 0 \vee k = 0 \\
qT_q(n-1, n-1) + T_q(n, n-1) & \textrm{if } n = k > 0 \\
T_q(n, k-1) + T_q(n-1,...
- 7,226
4
votes
1
answer
130
views
Intersecting algorithm for A065601
Let $a(n)$ be A065601 (i.e., number of Dyck paths of length $2n$ with exactly $1$ hill). Here
$$
a(n) = \frac{1}{2(n+1)}((3n-2)a(n-1) + 2(9n-19)a(n-2) + 4(2n-3)a(n-3)), \\
a(0) = a(2) = 0, a(1) = 1.
$$...
- 34.3k
4
votes
1
answer
112
views
On a number of compositions of $n$ into positive triangular numbers
Let $a(n)$ be A023361 (i.e., number of compositions of $n$ into positive triangular numbers). Here
$$
a(n) = \sum\limits_{i \geqslant 1, \frac{i(i+1)}{2}\leqslant n} a(n-\frac{i(i+1)}{2}), \\
a(0) = 1....
- 34.3k
3
votes
1
answer
178
views
Algorithm for the sum with binomial coefficients and Bell numbers
Let $a(n)$ be A000110 (i.e., Bell or exponential numbers: number of ways to partition a set of $n$ labeled elements).
Let $b(n)$ be A355247 (i.e., expansion of exponential generating function $\exp(2(\...
- 34.3k
2
votes
0
answers
46
views
On A088352 as an antidiagonal sums of A129179
Let $a(n)$ be A088352. Here $a(n)$ is an integer sequence with generating function $A(x)$ such that
$$
A(x) = \cfrac{1}{1-x-\cfrac{x^2}{1-x^3-\cfrac{x^4}{1-x^5-\cfrac{x^6}{1-x^7-\cfrac{x^8}{\ddots}}}}}...
- 12.9k
3
votes
2
answers
511
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 ...
- 2,836
7
votes
4
answers
793
views
Must bounded sequences be well-distributed to most *composite* moduli?
Let $\{a_n\}_{n=1}^N$, $|a_n|\leq 1$. Let $Q=\sqrt{N}$. Then $a_n$ is well-distributed modulo most prime $p\leq Q$, in the following sense:
$$\sum_{p\leq Q} \frac{1}{p} \left(\frac{1}{N/p} \sum_{\...
- 148k
2
votes
1
answer
310
views
Generating function for A300483 (related to Chebyshev polynomial of first kind)
Let $a(n)$ be A300483. Here
$$
a(n) = 2\int\limits_{t \geqslant 0}T_n\left(\frac{t+1}{2}\right)\exp(-t)\,dt.
$$
where $T_n(x)$ is $n$-th Chebyshev polynomial of first kind.
Let $b(n)$ be an integer ...
- 34.3k
10
votes
1
answer
625
views
Generating function for A261041
Let $a(n)$ be A261041 (i.e., number of partitions of subsets of $\{1,2,\dotsc,n\}$, where consecutive integers are required to be in different parts).
Let $b(n)$ be an integer sequence with generating ...
- 34.3k
2
votes
2
answers
315
views
5 different ways to define the same family of integer sequences
Let ${n \brace k}$ be a Stirling number of the second kind.
Let $A_n(x)$ be an Eulerian polynomial. Here
$$
A_n(x) = \sum_{i=0}^{n}i!{n \brace i}(x-1)^{n-i}.
$$
Let $a_1(n,p,q)$ be the family of ...
- 17k
2
votes
0
answers
157
views
What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
- 4,829
1
vote
1
answer
101
views
Finding $\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma$ from the generating function of $\zeta(•)$
In equation (130) of this page, the identity $$\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma \label{1} \tag{1} $$ is stated. Here, $\zeta(\cdot)$ is ...
- 105k
1
vote
0
answers
105
views
Simpler recursion for the A358612
Let $T(n,k)$ be an integer coefficients (A358612) such that
$$
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
$$
...
- 175
3
votes
1
answer
140
views
$R$-recursion for unsigned Genocchi numbers (of first kind) of even index
Let $G_n$ be A036968 (i.e., Genocchi numbers). Here
$$
\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}\...
- 175
6
votes
1
answer
282
views
Integer sequences with a periodic pattern
Let $A$ and $B$ be two different integers. Let $S$ be a finite integer sequence with exactly $n_A$ $A$s and $n_B$ $B$s. By repeating $S$ infinitely many times we obtain an infinite integer sequence $P$...
- 39.7k
7
votes
2
answers
265
views
On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?
I. Recurrences
In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation,
$$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$
within a ...
- 14.6k
6
votes
1
answer
387
views
Lucas number multiples of Fibonacci pairs
$\newcommand{\GCD}{\operatorname{GCD}}$ For $n=0,1,2,\ldots,$ let $F_n=0,1,1,2,3,5,\ldots$ and $L_n=2,1,3,4,7,11,\ldots$ be the Fibonacci and Lucas sequences. I expect the following is well known, but ...
3
votes
0
answers
165
views
Elegant algorithm for A140717
Let $T(n, k)$ be A140717 (i.e., triangle read by rows: $T(n,k)$ is the number of Dyck paths $d$ of semilength $n$ such that sum of peakheights of $d$ - number of peaks of $d$ equals $k$ ($n \geqslant ...
- 4,938
1
vote
0
answers
86
views
Decouple two continued-fractions
Given a three-term recurrence relation such as
\begin{equation}
a_nX^n+b_nX^{n+1}+c_nX^{n-1}=\zeta_n \delta_{n,0}
\end{equation}
it can always be written in continued-fraction form, and if you are ...
- 19
2
votes
0
answers
64
views
On a $\sum\limits_{n=0}^{\infty}c_n x^n=\sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n+1}(1-f(k)x^k)$ (slightly different question)
Please note that this question differs from one of the previous questions of mine.
Let $f(n)$ be an arbitrary function with integer values.
Let $c_n$ be an arbitrary integer sequence.
Let $a(n)$ be ...
- 4,938
1
vote
0
answers
32
views
On a A347205 and related row polynomials
Let $a(n)$ be A347205. Here
$$
a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^j k), \\
a(0) = 1.
$$
Let $\nu_2(n)$ be A007814 (i.e., number of trailing zeros in the binary expansion of $n$). Here
$$
\nu_2(2n+...
- 4,938
3
votes
0
answers
161
views
On generators of the multiplicative semigroup $\{r\in\mathbb Q:\ r>1\}$
The set $M=\{r\in\mathbb Q:\ r>1\}$ is a commutative semigroup with respect to the multiplication. For any integers $a>b\ge1$, we clearly have
$$\frac ab=\prod_{n=b}^{a-1}\frac{n+1}n.$$
So the ...
- 15.6k