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Tagged with catalan-numbers generating-functions
7 questions
5
votes
1
answer
148
views
Identities for the generating functions of a sort of convolution powers of the Narayana numbers
Let $c(x)=\frac{1-\sqrt{1-4x}}{2x}$ be the generating function of the Catalan numbers.
It satisfies $$\frac{1}{c(x)^k}+x^k c(x)^k=L_k(1,-x),$$
where $L_n(x,s)$ denote the Lucas polynomials defined by $...
- 5,612
4
votes
1
answer
423
views
Generating functions for Hankel determinants of Catalan numbers
The Hankel determinants of the Catalan numbers are well known and can be written as
$d(k,n)= \det \left( C_{k + i + j} \right)_{i,j = 0}^{n - 1}=\prod_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$...
- 5,612
15
votes
1
answer
259
views
A formula for this generating function that is similar to the $qt$-Catalan numbers
I came up with the following conjecture:
$$
\sum_{n \ge 0} z^n \sum_{\mu \vdash n} \frac{ t^{\sum l}q^{\sum a}}{\prod (q^a - t^{l+1})(t^l - q^{a+1})} = \exp\left(\sum_{n \ge 1} \frac{z^n}{n(q^n-1)(t^n-...
- 1,509
11
votes
2
answers
1k
views
Proofs of some combinatorial identities
Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-...
- 1,190
7
votes
0
answers
530
views
Can we define the Mandelbrot set in terms of the generating function of the Catalan numbers?
For each complex number $c$, define $P_{0}(c)=0$ and $P_{n+1}(c) = (P_{n}(c))^{2} + c$ . The Mandelbrot set is the set of complex numbers c for which $|P_{n}(c)|$ stays bounded as $n\rightarrow \...
1
vote
2
answers
193
views
Solving for f given constraint involving f(x, y) and f(xy, y)
I am interested in a weighted version of the Catalan numbers. The generating function for this case,
$$ f(x, y) = \sum_s \sum_n f_{s n} x^s y^n $$
(where the $y^n$ term is the weight), obeys the ...
- 403
4
votes
0
answers
144
views
Using Mellin transform for a certain function
In short, I want to use the Mellin transform to obtain the asymptotic behavior of the sequence $D_n = \frac{ [z^n] D(z)} {C_n}$ where
$$
D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( \sqrt{1-4z+4z^{p+...
- 141