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Tagged with catalan-numbers sequences-and-series
6 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
$...
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, $g(2n) = n$...
0
votes
4
answers
570
views
How to compute this series: $\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$ [closed]
How to compute this series: $$\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$$ where $C_k$ is the catalan number: $C_k=\frac{1}{k+1}{2k \choose k}$. (Further, is there any general method to treat this ...
5
votes
1
answer
340
views
Show a sequence of sums involving Catalan Numbers converges
Let $C_n$ be the $n$-th Catalan Number and let $\mathcal{O}_{s,j} = {{2s-j-1}\choose{j}} C_{s-j}^2$. Then we want to consider $\mathcal{E}_s = \sum_{j=0}^{s-1} (-1)^j\mathcal{O}_{s,j}$. We want to ...
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....
1
vote
2
answers
193
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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 ...