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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.
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%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