%I #16 Jan 09 2022 23:50:20
%S 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,
%T 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
%N 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.
%C For n >= 1, row sums are given by A152061.
%C Conjecture: T(n,n+1) = 1 if and only if n is a Mersenne prime (A000668).
%C Conjecture: T(2*n,2) = n.
%C Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1.
%e n\k| 1 2 3 4 5 6 7 8 9 10 11
%e ---+----------------------------------
%e 0 | 1
%e 1 | 0, 0
%e 2 | 1, 1, 0
%e 3 | 1, 0, 0, 1
%e 4 | 1, 2, 1, 0, 0
%e 5 | 1, 0, 0, 1, 0, 0
%e 6 | 1, 3, 3, 2, 1, 0, 0
%e 7 | 1, 0, 0, 0, 0, 0, 0, 1
%e 8 | 1, 4, 6, 4, 1, 0, 0, 0, 0
%e 9 | 1, 0, 0, 4, 1, 0, 2, 0, 0, 0
%e 10 | 1, 5, 10, 11, 5, 1, 0, 0, 1, 0, 0
%e The T(6,4) = 2 degree-6 polynomials over Z/2Z with k=4 nonzero terms are
%e 1 + x^2 + x^4 + x^6 = (1 + x^2)^3 = (1 + x + x^2 + x^3)^2, and
%e x^3 + x^4 + x^5 + x^6 = (x + x^2)^3.
%Y Cf. A000668, A152061.
%K nonn,tabl
%O 0,12
%A _Peter Kagey_, Jan 03 2022