A350532 Triangle read by rows: T(n,k) is the number of degree-n polynomials over Z/2Z of the form f(x)^m for some m > 1 with exactly k nonzero terms; 1 <= k <= n + 1.

1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 3, 3, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 4, 6, 4, 1, 0, 0, 0, 0, 1, 0, 0, 4, 1, 0, 2, 0, 0, 0, 1, 5, 10, 11, 5, 1, 0, 0, 1, 0, 0

Offset: 0

Author

Peter Kagey, Jan 03 2022

Keywords

• nonn
• tabl

Comments

For n >= 1, row sums are given by A152061.

Conjecture: T(n,n+1) = 1 if and only if n is a Mersenne prime (A000668).

Conjecture: T(2*n,2) = n.

Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1.

Examples

			  n\k| 1  2   3   4  5  6  7  8  9 10 11
  ---+----------------------------------
   0 | 1
   1 | 0, 0
   2 | 1, 1,  0
   3 | 1, 0,  0,  1
   4 | 1, 2,  1,  0, 0
   5 | 1, 0,  0,  1, 0, 0
   6 | 1, 3,  3,  2, 1, 0, 0
   7 | 1, 0,  0,  0, 0, 0, 0, 1
   8 | 1, 4,  6,  4, 1, 0, 0, 0, 0
   9 | 1, 0,  0,  4, 1, 0, 2, 0, 0, 0
  10 | 1, 5, 10, 11, 5, 1, 0, 0, 1, 0, 0
The T(6,4) = 2 degree-6 polynomials over Z/2Z with k=4 nonzero terms are
1 + x^2 + x^4 + x^6 = (1 + x^2)^3 = (1 + x + x^2 + x^3)^2, and
x^3 + x^4 + x^5 + x^6 = (x + x^2)^3.
		

Crossrefs

Cf. A000668, A152061.