Written: March 23, 2021

Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org

Last update: June 17, 2025

Maple packages

• StanleyStern.txt, a Maple package to automatically generate generating functions for generalized Stern sums conisered by Richard Stanley Stern's diaatomic array and generalized versions of it.

• 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.

Sample Input and Output Files for StanleyStern.txt

• If you want to see a computer-generated book with generating functions for the sums of the r^th power of the rows in the ORIGINAL Stern diatomic array, for r from 1 to 15 considered by Richard Stanley, using the NON-EXPERIMENTAL approach, that solves a linear system equation with SYMBOLIC coefficients
  The input gives you the output   .

• If you want to see a computer-generated book with generating functions for the sums of the r^th power of the rows in the ORIGINAL Stern diatomic array, for r from 1 to 15 considered by Richard Stanley, using the EXPERIMENTAL (yet rigorous!) approach, that solves a linear system of equation with NUMERIC coefficients
  The input gives you the output   .

• Since the latter approach is much faster, we can easily get the generating functions for r from 1 to 25
  The input gives you the output   .

• If you want to see a computer-generated book with generating functions for the sums of products of r consecutive terms of the rows in the ORIGINAL Stern diatomic array, for r from 1 to 9 considered by Richard Stanley, using the NON-EXPERIMENTAL approach
  The input gives you the output   .

• If you want to see a computer-generated book with generating functions for the sums of products of r consecutive terms of the rows in the ORIGINAL Stern diatomic array, for r from 1 to 9 considered by Richard Stanley, using the (faster, in this case, but not that fast), EXPERIMENTAL approach
  The input gives you the output.   .

Sample Input and Output Files for StanleyCF.txt

• If you want to see generating functions for the sum of the r-th power of the coefficients of the polynomial

  Prod(1 + x^{F_{i + 1}} + x^F_i+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   .

• Letting

  Sum( a(n,k) x^k ,k=0..infinity)=Prod(1 + x^{F_{i + 1}} + x^F_i+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   .

• Let T_i be the Tribonacci numbers (defined by T₁=1 ,T₂=1, ,T₃=1, and for i ≥ 4

  T_i = T_i-1 +T_i-2 + T_i-3  ,

  Let

  Sum( a(n,k) x^k ,k=0..infinity)=Prod(1 + x^{T_{i + 1}} + x^T_i+2 + x^T_i+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.   .

• Since things are getting complicated, for r=4, we had to resort to just getting the huge matrix, and enables fast computation of the sequences. So if you want to see a matrix version of the above, including r=4 Let T_i be the Tribonacci numbers, and let
  The input gives you the output   .

  Note that the matrix for the r=4 case has dimension 7245

• As above, let T_i be the Tribonacci numbers, and let

  Sum( a(n,k) x^k ,k=0..infinity)=Prod(1 + x^{T_{i + 1}} + x^T_i+2 + x^T_i+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   .

• If you want to see the first few terms (by computing the matrix, but not solving the system) for

  Sum(a(n,k)a(n,k+1)a(n,k+2),k=0..infinity)

  The input gives you the output   .

• Let Q_i be the Quadronacci numbers (defined by Q₁=1 ,Q₂=1 , Q₃=1, , Q₄=1, and for i ≥ 5

  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^{Q_{i + 1}} + x^Q_i+2 + x^Q_i+3 + x^Q_i+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   .

• Again, let Q_i be the Quadronacci numbers (defined by Q₁=1 ,Q₂=1 , Q₃=1, , Q₄=1, and for i ≥ 5

  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^{Q_{i + 1}} + x^Q_i+2 + x^Q_i+3 + x^Q_i+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   .

• If you want to see a more succinct version of the previous file, just stating the dimension of the matrix and giving the first 30 terms

  The input gives you the output   .

• Let P_i be the Pentaronacci numbers (defined by P₁=1 ,P₂=1 , P₃=1, , P₄=1, , P₅=1, and for i ≥ 6

  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^{P_{i + 1}} + x^P_i+2 + x^P_i+3 + x^P_i+4 + x^P_i+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   .

• If you want to see a table of the generating functions J_r^(k)(t,x) defined in Richard Stanley's article (arXiv:2101.02131v2) for r from 2 to 10 and k from 2 to 4,

  The input gives you the output   .

• If you want to see a table of the generating functions J_r^(k)(1,x) defined in Richard Stanley's article (arXiv:2101.02131v2) for r from 2 to 20 and GENERAL (SYMBOLIC!) k [conjectured and verified for k=2,3,4]

  The input gives you the output   .

• If you want to see a the first 26 terms in the (most probably) very hard-to-compute sequnce for the sum of squares of the coefficeints of Product(1+x^(2^i+(-1)^i)+x^(2^(i+1)+(-1)^(i+1)),i=0..n-1)

  The input gives you the output   .

• [Added March 30, 2021] If you want to see generating functions for the sum of the r-th power of the coefficients of Prod(1+x^f(i+2),i=1..n) where f(i) is a solution of the recurrence f(i)=f(i-1)+f(i-3) that (with initial conditions f(1)=f(2)=f(3)=1) Richard Stanley on the top of p. 21 of his
  article (arXiv:2101.02131v2)
  doubted is "nice", for r=1,2,3,4

  The input gives you the output   .

• [Added March 30, 2021] If you want to see generating functions for the sum of a(n,k)(a,k+1)...a(n,k+r-1) over k, where a(n,k) are the coefficients of Prod(1+x^f(i+2),i=1..n) where f(i) is a solution of the recurrence f(i)=f(i-1)+f(i-3), f(1)=f(2)=f(3)=1 for r=1,2,3

  The input gives you the output   .

• [Added March 31, 2021] If you want to see generating functions for the sum of the r-th power of the coefficients of Prod(1+x^f(i+2),i=1..n) where f(i) is a solution of the recurrence f(i)=f(i-2)+f(i-3) that Richard Stanley on the top of p. 21 of his
  article (arXiv:2101.02131v2)
  doubted is "nice", for r=1,2,3

  The input gives you the output   .

• [Added March 31, 2021] If you want to see generating functions for the sum of a(n,k)(a,k+1)...a(n,k+r-1) over k, where a(n,k) are the coefficients of Prod(1+x^f(i+2),i=1..n) where f(i) is a solution of the recurrence f(i)=f(i-2)+f(i-3) for r=1,2

  The input gives you the output   .

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