All Questions
Tagged with catalan-numbers nt.number-theory
12 questions
2
votes
1
answer
147
views
$R$-recursion for Fibonacci numbers using signed Catalan numbers
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1.
$$
Let $C_n$ be A000108 (i.e., Catalan numbers). Here
$$
C_n = \frac{1}{n+1}\binom{2n}{n}.
$$
Let
$...
- 4,928
6
votes
1
answer
281
views
Another generalization of parity of Catalan numbers
Recently, a question by T. Amdeberhan gathered up many enjoyable proofs that a Catalan number $C_n$ is odd if and only if $n=2^r-1$.
Noam D. Elkies' answer considered $F=\sum_{n=0}^\infty C_n x^{n+1}$....
- 524
3
votes
1
answer
193
views
Sequences that sums up to second differences of Bell and Catalan numbers
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Let $g(n)$ be A025480, $g(2n) = n$...
- 4,928
1
vote
1
answer
257
views
A divisibility problem involving Catalan numbers
The Catalan numbers in combinatorics are given by
$$C_n=\frac1{n+1}\binom{2n}n=\binom{2n}n-\binom{2n}{n+1}\ \ (n=0,1,2,\ldots).$$
In 2014 I formulated the following conjecture.
Conjecture. For each $...
- 15.6k
3
votes
1
answer
224
views
"Oddity" of $q$-Catalan polynomials: Part II
This question extends my earlier MO post for which I'm grateful for answers and useful comments.
The Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$ satisfy:
$\text{$C_{1,n}$ is odd iff $n=2^j-1$ for ...
- 43.1k
9
votes
2
answers
709
views
Oddity of generalized Catalan numbers: Part I
The famous (classical) Catalan numbers $C_{1,n}=\frac1{n+1}\binom{2n}n$ satisfy the following well-known arithmetic property:
$$\text{$C_{1,n}$ is odd iff $n=2^j-1$ for some $j$}.\tag1$$
Consider the "...
- 43.1k
9
votes
1
answer
470
views
Products of Catalan numbers
Let $c(n)=\frac{1}{n+1}\binom{2n}{n}$ be the Catalan number. It seems that a product $\prod_{n\in I} c(n)$, where $I\subset\mathbb N_{>1}$, is never a Catalan number. Is this a (known) fact?
- 5,792
5
votes
1
answer
613
views
generating $q$-Catalan numbers
An $n$-Dyck path (or a Catalan path) is a lattice path $P$, unit East and North steps, in an $n\times n$ square grid which stays (weakly) above the main diagonal. Let $\square_n$ denote all such paths....
- 43.1k
14
votes
1
answer
2k
views
When does a Catalan number equal a Fibonacci number?
The $n=3$'rd Catalan number (A000108) is $1,1,2,5$ : $\frac{\binom{2n}{n}}{n+1}=\frac{\binom{6}{3}}{4}=\frac{20}{4}=$ 5.
The $n=4$'th Fibonacci number (A000045) is $1,1,2,3,5,...$ : 5.
Q. Which ...
- 151k
9
votes
3
answers
1k
views
Intuition Behind a Decimal Representation with Catalan Numbers
From $0 = 0.5 - 0.5 = 0.5 - \sqrt{0.25}$, we can adjust the subtrahend slightly to obtain
$$0.5 - \sqrt{0.249} = 0.001\ 001\ 002\ 005\ 014\ 042\ldots$$
where the decimal representation contains the ...
- 7,831
7
votes
4
answers
3k
views
A generalization of Catalan numbers
It is well-known that the $n$th Catalan number is equal to $(n+1)^{-1}\binom{2n}{n}$. A long time ago I had wondered what happens if you look at the sequence generated by $(n+k)^{-1}\binom{pn}{n}$ - ...
- 6,962
8
votes
2
answers
810
views
Modular congruences related to sums of Catalan numbers
I am curious if somebody can be helpful concerning the following
experimental observation:
There exist two rational sequences $\alpha_0,\alpha_1,\dots$ and
$\beta_0,\beta_1,\dots$, both with values ...
- 17.9k