Mixing
Nonlinear Recursion Solution Process for $x_{n+1}=Sigma_{i=1}^{n} x_{i}x_{n-i}$ (Known Solution)
0
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I want to solve the equation $x_{n+1}=Sigma_{i=1}^{n} x_{i}x_{n-i}$. Plugging the equation into Mathematica gives me $x_n=(-1)^{n}2^{2n+1} Binomial(1/2, n+1)x_0^{n+1}$. How might I derive this?
recurrence-relations recursion quadratic-forms nonlinear-system
asked Jan 18 at 17:23
BJParksBJParks
296
$endgroup$
add a comment |
0
$begingroup$
I want to solve the equation $x_{n+1}=Sigma_{i=1}^{n} x_{i}x_{n-i}$. Plugging the equation into Mathematica gives me $x_n=(-1)^{n}2^{2n+1} Binomial(1/2, n+1)x_0^{n+1}$. How might I derive this?
recurrence-relations recursion quadratic-forms nonlinear-system
asked Jan 18 at 17:23
BJParksBJParks
296
$endgroup$
1
$begingroup$
With generating functions?
$endgroup$
– Somos
Jan 18 at 17:52
$begingroup$
@Somos I'm not entirely sure how to use generating functions, but thank you, I will see if I can read about them and understand how to use them.
$endgroup$
– BJParks
Jan 18 at 18:15
2
$begingroup$
See Catalan number first proof for an example.
$endgroup$
– Somos
Jan 18 at 18:17
$begingroup$
@Somos Thank you! That's exactly what I'm looking for, and it really helps me learn the process so I can try extending it to other situations. Thanks so much!
$endgroup$
– BJParks
Jan 18 at 18:27
add a comment |
0
0
0
$begingroup$
I want to solve the equation $x_{n+1}=Sigma_{i=1}^{n} x_{i}x_{n-i}$. Plugging the equation into Mathematica gives me $x_n=(-1)^{n}2^{2n+1} Binomial(1/2, n+1)x_0^{n+1}$. How might I derive this?
recurrence-relations recursion quadratic-forms nonlinear-system
asked Jan 18 at 17:23
BJParksBJParks
296
$endgroup$
I want to solve the equation $x_{n+1}=Sigma_{i=1}^{n} x_{i}x_{n-i}$. Plugging the equation into Mathematica gives me $x_n=(-1)^{n}2^{2n+1} Binomial(1/2, n+1)x_0^{n+1}$. How might I derive this?
recurrence-relations recursion quadratic-forms nonlinear-system
recurrence-relations recursion quadratic-forms nonlinear-system
asked Jan 18 at 17:23
BJParksBJParks
296
asked Jan 18 at 17:23
BJParksBJParks
296
asked Jan 18 at 17:23
BJParksBJParks
296
asked Jan 18 at 17:23
BJParksBJParks
296
asked Jan 18 at 17:23
BJParksBJParks
296
296
1
$begingroup$
With generating functions?
$endgroup$
– Somos
Jan 18 at 17:52
$begingroup$
@Somos I'm not entirely sure how to use generating functions, but thank you, I will see if I can read about them and understand how to use them.
$endgroup$
– BJParks
Jan 18 at 18:15
2
$begingroup$
See Catalan number first proof for an example.
$endgroup$
– Somos
Jan 18 at 18:17
$begingroup$
@Somos Thank you! That's exactly what I'm looking for, and it really helps me learn the process so I can try extending it to other situations. Thanks so much!
$endgroup$
– BJParks
Jan 18 at 18:27
add a comment |
1
$begingroup$
With generating functions?
$endgroup$
– Somos
Jan 18 at 17:52
$begingroup$
@Somos I'm not entirely sure how to use generating functions, but thank you, I will see if I can read about them and understand how to use them.
$endgroup$
– BJParks
Jan 18 at 18:15
2
$begingroup$
See Catalan number first proof for an example.
$endgroup$
– Somos
Jan 18 at 18:17
$begingroup$
@Somos Thank you! That's exactly what I'm looking for, and it really helps me learn the process so I can try extending it to other situations. Thanks so much!
$endgroup$
– BJParks
Jan 18 at 18:27
1
1
$begingroup$
With generating functions?
$endgroup$
– Somos
Jan 18 at 17:52
$begingroup$
With generating functions?
$endgroup$
– Somos
Jan 18 at 17:52
$begingroup$
@Somos I'm not entirely sure how to use generating functions, but thank you, I will see if I can read about them and understand how to use them.
$endgroup$
– BJParks
Jan 18 at 18:15
$begingroup$
@Somos I'm not entirely sure how to use generating functions, but thank you, I will see if I can read about them and understand how to use them.
$endgroup$
– BJParks
Jan 18 at 18:15
2
2
$begingroup$
See Catalan number first proof for an example.
$endgroup$
– Somos
Jan 18 at 18:17
$begingroup$
See Catalan number first proof for an example.
$endgroup$
– Somos
Jan 18 at 18:17
$begingroup$
@Somos Thank you! That's exactly what I'm looking for, and it really helps me learn the process so I can try extending it to other situations. Thanks so much!
$endgroup$
– BJParks
Jan 18 at 18:27
$begingroup$
@Somos Thank you! That's exactly what I'm looking for, and it really helps me learn the process so I can try extending it to other situations. Thanks so much!
$endgroup$
– BJParks
Jan 18 at 18:27
add a comment |
0
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1
$begingroup$
With generating functions?
$endgroup$
– Somos
Jan 18 at 17:52
$begingroup$
@Somos I'm not entirely sure how to use generating functions, but thank you, I will see if I can read about them and understand how to use them.
$endgroup$
– BJParks
Jan 18 at 18:15
2
$begingroup$
See Catalan number first proof for an example.
$endgroup$
– Somos
Jan 18 at 18:17
$begingroup$
@Somos Thank you! That's exactly what I'm looking for, and it really helps me learn the process so I can try extending it to other situations. Thanks so much!
$endgroup$
– BJParks
Jan 18 at 18:27