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Results tagged with sequences-and-series
Search options questions only
not deleted
user 66131
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.
8
votes
1
answer
629
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 Fibo …
- 79.9k
4
votes
1
answer
128
views
Solving a three-parameter recursive sequence
Consider the triple-indexed sequence of integers defined by
\begin{align} \label{coefficientsV} \nonumber
f(\alpha,\beta,\gamma)
&:=(2\alpha+8\beta+12\gamma-1)\cdot f(\alpha-1,\beta,\gamma) - …
- 3,791
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?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \prod_{n\geq1}(1-x^{ …
- 105k
1
vote
0
answers
162
views
Triangular and pentagonal numbers in $q$-series
Consider the following two infinite series
$$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\,
\sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2 …
- 43.2k
4
votes
1
answer
300
views
3 divides coefficents of this $q$-series
Denote $\phi(q):=\prod_{j\geq1}(1-q^j)$ and let $\xi=e^{\frac{2\pi i}3}$ be a cube root of unity.
Define the sequence $u(n)$ by
$$\prod_{n\geq1}\prod_{s=1}^2(1-q^n\xi^{ns})(1-q^{2n}\xi^{ns})
=\sum_{n\ …
- 5,624
6
votes
2
answers
717
views
Recreation with Catalan
Consider the well-known sequence $C_k=\frac1{k+1}\binom{2k}k$ of Catalan numbers. I came across the below identity while working with certain generating functions. I thought it might be of interest to …
- 17k
0
votes
1
answer
345
views
A combinatorial proof: where art thou?
Start by introducing the finite sums
$$A_n:=\sum_{m=1}^nq^m\prod_{j=1}^{m-1}(1-q^j) \qquad \text{and} \qquad
B_n:=\sum_{m=1}^nq^m\prod_{j=m+1}^n(1-q^j).$$
An algebraic proof is facile: Clearly, $A_1=B …
- 109k
3
votes
1
answer
156
views
$q$-series and Stirling of the 1st kind
Denote the (unsigned) Stirling numbers of the $1^{st}$-kind by ${n \brack k}$ and define
$$\mathbf{F}_a(q)=\sum_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad
\mathbf{G}_b(q)=\sum_{m\geq …
- 7,226
6
votes
3
answers
527
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 …
- 10.5k
1
vote
0
answers
142
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 th …
- 43.2k
1
vote
1
answer
340
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 infi …
- 43.2k
46
votes
5
answers
4k
views
Fibonacci series captures Euler $e=2.718\dots$
The Fibonacci recurrence $F_n=F_{n-1}+F_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question m …
- 2,784
6
votes
1
answer
258
views
are endomorphisms "small" compared to the full transformations?
$\DeclareMathOperator\End{End}$Let $T_n$ be the full transformation semigroup/monoid of $[n]=\{1,\dots,n\}$. Let $\End(T_n)$ be the set of [endomorphisms][1] of $T_n$. Then, $\# T_n=n^n$ and
$$\# \End …
- 63.9k
7
votes
3
answers
735
views
Expanding in Fibonacci powers
Let $F_n$ denote the all-familiar Fibonacci numbers, with $F_0=0, F_1=1, F_2=1$, etc.
There is a plethora of properties for these numbers involving their sums, products, convolutions and so on. Here, …
CommunityBot
- 1
1
vote
0
answers
86
views
Doubly log-concave or doubly log-convex
Suppose $(a_k)_{k\geq0}$ is a sequence of real numbers. Consider the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$.
We say $(a_k)_k$ is log-concave (resp. log-convex) provided $\mathcal{L}a_k\geq0$ ( …
- 43.2k