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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,12
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.
LINKS
EXAMPLE
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
Sequence in context: A107782 A368413 A086017 * A000161 A060398 A253242
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Jan 03 2022
STATUS
approved

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Last modified July 13 12:36 EDT 2024. Contains 374284 sequences. (Running on oeis4.)