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Results tagged with recurrences
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user 168671
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Is the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n!$ and $f_{m,n} = f_{m-1,n}+ f_{m,n...
Is the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n!$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$ P-recursive?
https://en.wikipedia.org/wiki/P-recursive_equation
https://en.wikipedia.org/ …
1
vote
0
answers
193
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Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n)} =\lim_{n\rightarrow \infty}\frac{F(n+...
Let $\alpha,\beta, \gamma \in \mathbb{R}^+$ be and the function
$$ F(m,n)= \begin{cases}
1, & \text{if $m n=0$ }; \\
\alpha F(m ,n-1)+ \beta F(m-1,n )+ \gamma F(m-1,n-1), & \text{ if $m n>0$. }%
\en …
3
votes
1
answer
305
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A generating function related to the Delannoy numbers
What is the generating function of $f_{m,n}$?
$ f_{m,n} = \begin{cases} 0 , & \text{if $m<0 $ or $ n<0$ }; \\
f_{n,m} , & \text{ if $n<m$}; \\
1, & \text{ if $0=m$ and $ n\in\{0,1\} $}; \\
…
-4
votes
1
answer
301
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Limit of recursion relation
Consider the sequence of functions $\{F_n\}_{n\in \mathbb{N}}$, where each $F_n$ is defined on $\{0,...,n\}$ by recurrence of the following form: $$ F_n(0)=3 \textrm{
and }F_n(k)=\frac{1}{k^2}+\frac{ …
1
vote
1
answer
342
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Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\frac{9}{8}$?
$$F(m,n)= \begin{cases}
1, & \text{if $m n=0$ }; \\
\frac{1}{2} F(m ,n-1) + \frac{1}{3} F(m-1,n )+ \frac{1}{4} F(m-1,n-1), & \text{ if $m n>0$. }%
\end{cases}$$
Please a proof of:
$$\lim_{n\rightarro …
3
votes
1
answer
144
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About the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_...
Let the sequence:
$s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$, for $mn>0$.
Computationally it seems that $\frac{s_{n+1}}{s_{n}} \approx e\cdot n - …
0
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1
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155
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How to probe the recursiveness order of a sequence $\{S_n\}$ whose generating function is known
How to probe the recursiveness order of a sequence $\{S_n\}$ whose generating function is known:
$$ \sum_{n\geq0} S_n z^n= \frac{4 z \left(\sqrt{49 z^2-18 z+1}+7 z-1\right)}{\sqrt{49
z^2-18 z+1} …
11
votes
3
answers
1k
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What is the limit of $a (n + 1) / a (n)$?
Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise.
What is the limit of $a(n + 1) / a (n)$? $(2.71...)$