Questions tagged [sequences-and-series]
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
1,731
questions
-1
votes
1
answer
120
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A permutation and combination problem about the number of connections in a sequence of n numbers [closed]
There is a sequence of n numbers as 1,2,3,...,n
How many combinations of the connections between two numbers in the sequence without overlaping?
...
0
votes
0
answers
51
views
Equivalence of recursions for A145879
Let $R_1(n,z)$ be row polynomials of A145879 i.e. of triangle read by rows: $T(n,k)$ is the number of permutations of $\left\lbrace 1,2,\cdots,n \right\rbrace$ having exactly $k$ entries that are ...
- 3,442
0
votes
0
answers
62
views
How to derive the formulas of the spin-weighted spheroidal eigenvalues (2.16a)-(2.16g) in arXiv:gr-qc/0511111?
I am reading the article "Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics
in four and higher dimensions", which is on https://arxiv.org/abs/gr-qc/0511111.
I want to ...
- 1
3
votes
1
answer
151
views
$q$-series and Stirling of the 1st kind
Denote the (unsigned) Stirling numbers of the $1^{st}$-kind by ${n \brack k}$ and define
$$\mathbf{F}_a(q)=\sum_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad
\mathbf{G}_b(q)=\sum_{m\...
- 41.8k
1
vote
0
answers
85
views
Suitable recursion for the A234289
Let $a(n)$ be A234289 i.e. integer sequence with exponential generating function
$$
A(x)=1+A(x)^2\int \frac{1}{A(x)}\,dx
$$
The sequence begins with
$$
1, 1, 3, 17, 147, 1729, 25827, 468593, 10012083, ...
- 3,442
1
vote
2
answers
151
views
Transcendental functions with two prescribed values
Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$.
I would like ...
- 515
1
vote
0
answers
95
views
Pretty simple recursion for the A290383
Let $a(n)$ be A290383 i.e. number of set partitions of $[n]$ such that the smallest element of each block is odd. Here
$$
a(n)=b(n,0,0)
$$
where
$$
b(n,m,t)=\sum\limits_{j=1}^{m-t+1}b(n-1,\max(m,j),1-...
- 3,442
4
votes
1
answer
247
views
Largest antichain in partial ordering in OEIS
OEIS A109388 $\{a_n\}_{n\ge1}$ is an integer sequence with $a_n=\binom{n}{\lfloor \frac{n}{3} \rfloor}\times 2^{n-\lfloor\frac{n}{3}\rfloor}$, I noticed that OEIS says
$a_n$ is the size of the ...
- 63
-3
votes
1
answer
160
views
Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value(V2)?
I have a question regarding this question here.
is-there-a-summation-method-where-the-divergent-series
if I set $ x+2=c/c-v$ , will I have
$U_n = M\left(c-\frac{c}{n+2}\right)-M\left(c-\frac{c}{n+1}\...
- 33
2
votes
0
answers
118
views
Extensions of Euler–Maclaurin formula
There are ways to approximate a sum through integration like the Euler–Maclaurin formula, which requires the function $f(x)$ to be continuous, but there are several ways to extend the formula to ...
- 67
1
vote
0
answers
77
views
Recursion for the A006014 using difference of binomial coefficients
Let $a(n)$ be A006014 i.e.
$$
a(n)=na(n-1)+\sum\limits_{j=1}^{n-2}a(j)a(n-j-1), \\
a(1)=1
$$
Also generating function $A(x)$ satisfies
$$
A(x) = x(1 + A(x) + A(x)^2 + xA'(x))
$$
Let
$$
R(n,q)=\sum\...
- 3,442
1
vote
1
answer
174
views
Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for $L^1_\text{loc}$ $f : [0,\infty) \to \mathbb{R}$?
Let $f(t) : [0,\infty) \to \mathbb{R}$ be an $L^1_\text{loc}$ function.
Then, I wonder if the following series
\begin{equation}
\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert
\, dt
...
- 2,749
0
votes
0
answers
68
views
Recursion for a given series reversion
Define the operator $\operatorname{SR}$, which is associated with the series reversion.
Let $a(n,m,k)$ be an integer sequence with generating function
$$
\frac{1}{x}\operatorname{SR}(x\frac{1-mx}{1-kx}...
- 3,442
4
votes
0
answers
118
views
Something (which might be called multi-continued fraction) for the A112487
Let $a(n)$ be A112487 i.e. an integer sequence with exponential generating function
$$
A(x)=\exp\left(\int (A(x)+A(x)^2)\,dx\right), \\
A(0)=1
$$
However, the definition in the name of the sequence is
...
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4
votes
1
answer
229
views
$\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}$ for various $x$
Let $$f(x)=\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}.$$
Compute $f(1)$ and $f(2)$.
0
votes
0
answers
100
views
Recursion for the A266328 by analogy with non-standard recursion for factorials
Let $a(n)$ be A266328 i.e. an integer sequence with exponential generating function
$$
A(x)=\exp\int B(x) \,dx
$$
such that
$$
B(x)=\exp(-x)\exp\int A(x) \,dx
$$
where the constant of integration is ...
- 3,442
7
votes
1
answer
747
views
Remarkable recursions for the A204262
Let $a(n)$ be A204262 i.e. permanent of the matrix $n\times n$ with elements $\min(i,j)$.
Let
$$
f_{n,\ell}(x)=g_{n,\ell}(x)+f_{n,\ell-1}(\ell)-g_{n,\ell}(\ell), \\
g_{n,\ell}(x)=\int (n-\ell)^2 f_{n-...
- 3,442
3
votes
1
answer
366
views
Convergence of a power series
Consider the numbers $$a_n=\frac{1}{n+1}\sum_{k=0}^{n}\frac{2^{k-1}\binom{n+1}{k}B_k}{2^{s+k-1}-1}, \ n\geq0,$$ where $s\neq1;0;-1;-2;-3;...$ is a fixed real number, and the $B_k$ are the Bernoulli ...
- 399
10
votes
2
answers
356
views
Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$
Consider the series
$$
\sum_{n=1}^{\infty} ( \{ n \xi \} - \frac{1}{2})
$$
where $\{ \}$ denotes the fractional part and $\xi$ is some irrational number. It clearly does not converge but can we show ...
- 177
1
vote
0
answers
104
views
Recursion for the Bessel polynomial $y_n(x)$
Let $a(n)$ be A001515 i.e. the Bessel polynomial $y_n(x)$ evaluated at $x=1$. Here
$$
a(n) = (2n-1)a(n-1) + a(n-2), \\
a(0) = 1, a(1) = 2
$$
The closed form is
$$
a(n)=\sum\limits_{k=0}^{n}\binom{n+k}{...
- 3,442
1
vote
1
answer
112
views
Wild's sum for Boltzmann's equation
Consider the spatially homogenous Boltzmann equation $$\partial_t f_t = Q^+(f_t,f_t) - f_t.$$ A semi-explicit representation formula for solutions of this Boltzmann equation can be written as (see for ...
- 700
0
votes
1
answer
144
views
Asymptotic of ratio between l1 / l2 norm of a structured vector
As suggested in this discussion, I would like to inquire about the following question:
Consider a matrix B of size $n\times n$ defined as:
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\...
- 335
0
votes
1
answer
110
views
asymptotic of ratio between two summations (l1 / l2 norm)
Let $B$ as a $n\times n$ matrix where
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\theta_j), 1\leq i<j\leq n$$ and other entries equals to $0$, and $$\theta=[\theta_1,\cdots,\theta_n]\...
- 335
0
votes
0
answers
78
views
Simple recursion for the A129179
Let $T(n,k)$ be A129179, i.e., an integer coefficient with generating function
$$
G(t,z) = 1 + zG(t,z) + tzG(t,t^2z)G(t,z)
$$
Other generating functions are $\frac{1}{G_1(t,z,0)}$ and $\frac{1}{G_2(t,...
- 3,442
3
votes
0
answers
156
views
Hilbert's 13th Problem and series solutions for the reduced sextic, septic, and octic?
I. Reduced equations
One can eliminate 3 terms from the general quintic, sextic, septic, and octic using a Bring-Jerrard transformation to get the reduced forms in radicals,
$$x^5+(x+p) = 0$$
$$x^6+(x+...
- 12.1k
2
votes
0
answers
70
views
Recursion for the number of partitions of $m^n-1$ into powers of $m$
Let $a(n,m)$ be the number of partitions of $m^n-1$ into powers of $m$. In other words,
$$a(n,m)=[z^{m^n-1}] \prod\limits_{k\geqslant 0} \frac{1}{1-z^{m^k}}$$
Let
$$
R(n,m,q)=\sum\limits_{j=0}^{m(q+1)-...
- 3,442
6
votes
3
answers
508
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 41.8k
1
vote
0
answers
89
views
Combinatorial interpretation for the more general case of $R(n,0)$
Let $f(n), g(n,m), h(n)$ be an arbitrary functions which equal to the non-negative integers.
Let
$$
R(n,q) = \sum\limits_{j=0}^{f(q)}g(q,j)R(n-1,j),\\
R(0,q) = h(q)
$$
In the comment to the one of ...
- 3,442
3
votes
1
answer
454
views
Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value?
Consider the function
$$
M(v) = \frac{m_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}},
$$ where $v \in \left]-c;c\right[$, $m_0\in\mathbb{R}^{*+}$, and $c=3\cdot10^8$.
Let $(U_n)$ be a sequence with ...
- 33
3
votes
1
answer
151
views
A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?
So I am wondering if there exists a general procedure for the following problem:
given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
- 2,283
13
votes
6
answers
2k
views
Closed form of an infinite series
Does the following infinite series have a closed form?
$$
\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}
$$
- 259
0
votes
1
answer
128
views
Series reversion using something like continued fraction
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let
$$
F(x)=\sum\limits_{m\geqslant 0}f(m)x^m
$$
Define the operator $\operatorname{SR}$, which is associated with the series ...
- 3,442
0
votes
0
answers
169
views
Expansion of continued fraction using recursion
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an integer sequence with generating function $\frac{1}{G(0)}$ where
$$
G(j)=1-\frac{f(j)x}{G(j+1)}
$$
Here we have
$$
G(...
- 3,442
0
votes
1
answer
193
views
Series involving sine and cosine
Let $(a_n)_n$ be an increasing real sequence with $a_n=O(\sqrt n)$.
Is it true that there exists an increasing function $\phi:\mathbb N\to\mathbb N$ such that $$\lim \left|\sum\limits_{k=1}^{\phi(n)}\...
- 3,795
1
vote
0
answers
88
views
Numerical strategies for evaluating a modular invariant infinite sum
I'm working on a problem that involves the numerical evaluation of the following infinite sum:
$$
\sum_{m=-\infty}^{\infty} \ln \left|1\pm e^{-2\pi \tau_1 \sqrt{m^2+x^2/(4\pi^2\tau_1)}-2 \pi i \tau_0 ...
- 11
9
votes
2
answers
357
views
Change of variable formulas in discrete calculus?
Crossposted from MSE.
In discrete calculus one defines the $h$-difference operator $$\Delta_h[f(x)] = f(x+h) - f(x)$$
and we often define $\Delta = \Delta_1.$ We can similarly define the indefinite ...
- 113
26
votes
1
answer
7k
views
Elegant recursion for A301897
Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here
$$a(n)=\frac{1}{n+1}\binom{2n}{...
- 3,442
7
votes
2
answers
371
views
A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
- 197
2
votes
0
answers
124
views
Recurrence for A004208
Let $a(n)$ be A004208. Here
$$a(n)=n\prod\limits_{j=1}^{n}(2j-1)-\sum\limits_{i=1}^{n-1}a(i)\prod\limits_{j=1}^{n-i}(2j-1)$$
I conjecture that
$$a(n)=R(n-1,0)$$
where
$$R(n,q)=2(q+2)R(n-1,q+1)+\sum\...
- 3,442
5
votes
1
answer
239
views
Operation preserving log-concavity of sequences
Here a log-concave sequence $(a_0,a_1,a_2,\ldots)$ is a sequence of positive real numbers such that $a_i^2 \geq a_{i-1}a_{i+1}$ for each $i\geq 1$. These are pervasive within mathematics.
A polynomial ...
- 1,879
0
votes
0
answers
63
views
General patterns for partial sums of generalized A341392, A284005 and A329369
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \operatorname{wt}(2n)=\operatorname{wt}(n), \operatorname{wt}(0)=0$$
$$T(n,k)=...
- 3,442
24
votes
1
answer
1k
views
Can we just use the linear term of exponential sums to sum divergent series
Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $
You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
- 2,283
6
votes
2
answers
314
views
Does control on the “magnitude” of the rearrangement give control of the rearranged Cesaro sums?
Let $a_n$ be a nonnegative sequence that Cesaro converges to $K > 0$. We recall this means
$$\frac{1}{N} \sum_{n = 1}^N a_n \to K$$
as $N \to \infty$.
Suppose $a_{\phi_n}$ with $\phi: \mathbb N \to ...
- 4,832
2
votes
0
answers
63
views
Recursive sequence of renewal type : when does one term dominate them all?
Let $(b_n)_{n \geq 0}$ be an increasing sequence of non negative real numbers.
Let $(u_n)_{n \geq 0}$ be recursively defined by $u_0 =1$ and
$$u_{n} = \sum_{k=0}^{n-1} u_{k} b_{n-k}$$
Find a ...
- 468
1
vote
0
answers
86
views
On level-$12$ of the McKay-Thompson series of the Monster and the Domb numbers
(This continues from level 10.) Given some moonshine functions $j_{N}$. There are nice descending and consistent relations for levels $6m$ with $m= 2,3,5,$
$$j_{12A} = \left(\sqrt{j_{12H}} + \frac{\...
- 12.1k
1
vote
0
answers
96
views
On level $6$ of the McKay–Thompson series of the Monster and Apéry numbers, et al
After the McKay–Thompson series of levels $1$, $2$, $3$, $4$ of the Monster were mentioned in this MO post, level $6$ has very interesting relations as well. (Level 10 is in this post.)
I. Level-6 ...
- 12.1k
1
vote
1
answer
185
views
Approximation for interpolation of harmonic numbers
I need a good approximations for $H_p$, for $p \in (0,1) \cap \mathbb{Q}$, the generalization of $H_n=\sum_{i=1}^n \frac{1}{i}$ to the real numbers.
I tried $H_p = p \sum_{k=1}^\infty \frac{1}{k (k + ...
- 125
2
votes
0
answers
198
views
Sum of roots of unity
Today I came across the series $\sum_{k=0}^{n-1}\varepsilon^{2k^2}$, where $\varepsilon$ is some primitive $n^\text{th}$ root of unity. Is there an explicit expression for this sum? I mistakenly ...
- 8,041
1
vote
0
answers
121
views
Hankel transform of certain $\pm1$ sequences
The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$
where $s_2(k)$ is ...
- 41.8k
1
vote
1
answer
332
views
Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 41.8k