By Shalosh B. Ekhad and Doron Zeilberger
Written: March 23, 2021
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org
SternCF.txt, A Maple package that studies C-finite generalizations of Stern's diatomic array inspired by the Fibonacci and k-bonacci generalizations considered by Richard Stanley.
Prod(1 + x^{Fi + 1} + x^{Fi+2} , i=1..n),
where F_{i} are the Fibonacci numbers,
for r from 1 to 6
[The case r=2 is given on line 3 of p. 10 of Richard Stanley's "Theorems and Conjectures on Some rational generating functions"]
The input gives you
the output .
[Of course, we used the NON-EXPERIMENTAL approach, since the experimental approach is hopeless]
For a version that only tells you the dimensions of the matrices, and uses it to get the first 30 terms
The input gives you
the output .
Sum( a(n,k) x^{k} ,k=0..infinity)=Prod(1 + x^{Fi + 1} + x^{Fi+2} , i=1..n)
If you want to see generating functions for the sequences
Sum(a(n,k)*a(n,k+1) , k=0..infinity) ,
Sum(a(n,k)*a(n,k+1)*a(n,k+2) , k=0..infinity) ,
Sum(a(n,k)*a(n,k+1)*a(n,k+2)*a(n,k+3) , k=0..infinity) ,
The input gives you the output .
T_{i} = T_{i-1} +T_{i-2} + T_{i-3} ,
Let
Sum( a(n,k) x^{k} ,k=0..infinity)=Prod(1 + x^{Ti + 1} + x^{Ti+2} + x^{Ti+3} , i=1..n)
If you want to see generating functions for the sequences
Sum(a(n,k)^{r},k=0..infinity) for r=1,2,3
The input gives you
the output. .
Note that the matrix for the r=4 case has dimension 7245
Sum( a(n,k) x^{k} ,k=0..infinity)=Prod(1 + x^{Ti + 1} + x^{Ti+2} + x^{Ti+3} , i=1..n)
If you want to see generating functions for the sequences
Sum(a(n,k)a(n,k+1),k=0..infinity)
Note that for
Sum(a(n,k)a(n,k+1)a(n,k+2),k=0..infinity)
it exceeded the number of equations that we allowed, but see below.
The input gives you the output .
Sum(a(n,k)a(n,k+1)a(n,k+2),k=0..infinity)
The input gives you the output .
Q_{i} = Q_{i-1} + Q_{i-2} + Q_{i-3} + Q_{i-4} , ,
Let
Sum( a(n,k) x^{k} ,k=0..infinity) = Prod(1 + x^{Qi + 1} + x^{Qi+2} + x^{Qi+3} + x^{Qi+4} , i=1..n)
If you want to see generating functions for the sequences
Sum(a(n,k)^{2} , k=0..infinity)
The input gives you the output .
Q_{i} = Q_{i-1} + Q_{i-2} + Q_{i-3} + Q_{i-4} , ,
and let
Sum( a(n,k) x^{k} ,k=0..infinity) = Prod(1 + x^{Qi + 1} + x^{Qi+2} + x^{Qi+3} + x^{Qi+4} , i=1..n)
If you want to see generating functions for the sequences
Sum(a(n,k)a(n,k+1) , k=0..infinity)
The input gives you the output .
P_{i} = P_{i-1} + P_{i-2} + P_{i-3} + P_{i-4} + P_{i-5} , ,
Let
Sum( a(n,k) x^{k} ,k=0..infinity) = Prod(1 + x^{Pi + 1} + x^{Pi+2} + x^{Pi+3} + x^{Pi+4} + x^{Pi+5} , i=1..n)
Then it is too complicated to get the generating function, since the matrix has dimension 12751, but we found the matrix, that enabled to compute the first few terms of
Sum(a(n,k)^{2} , k=0..infinity)
The input gives you the output .
The input gives you the output .
The input gives you the output .
The input gives you the output .
The input gives you the output .
The input gives you the output .
The input gives you the output .
The input gives you the output .