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A329721
Irregular triangular array T(n,k) read by rows: T(n,k) is the number of degree n monic polynomials in GF(2)[x] with exactly k distinct factors in its unique factorization into irreducible polynomials.
0
2, 3, 1, 4, 4, 6, 9, 1, 8, 20, 4, 14, 35, 15, 20, 70, 36, 2, 36, 122, 90, 8, 60, 226, 196, 30, 108, 410, 414, 91, 1, 188, 762, 848, 242, 8, 352, 1390, 1719, 601, 34, 632, 2616, 3406, 1416, 122, 1182, 4879, 6739, 3207, 374, 3, 2192, 9196, 13274, 7026, 1062, 18
OFFSET
1,1
COMMENTS
Observed row lengths are 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, ...
FORMULA
G.f.: Product_{k>=1} (y/(1-x^k) - y + 1)^A001037(k).
EXAMPLE
2;
3, 1;
4, 4;
6, 9, 1;
8, 20, 4;
14, 35, 15;
20, 70, 36, 2;
36, 122, 90, 8;
60, 226, 196, 30;
108, 410, 414, 91, 1;
...
T(5,3) = 4 because we have: x(x+1)(x^3+x+1), x(x+1)(x^3 +x^2+1), x^2(x+1)(x^2+x+1), x(x+1)^2(x^2+x+1).
MATHEMATICA
nn = 10; a = Table[1/m Sum[MoebiusMu[m/d] 2^d, {d, Divisors[m]}], {m, 1,
nn}]; Grid[Map[Select[#, # > 0 &] &, Drop[CoefficientList[Series[Product[(u/(1 - z^m ) - u + 1)^a[[m]], {m, 1, nn}], {z, 0, nn}], {z, u}], 1]]]
CROSSREFS
Row sums give A000079.
Column k=1 gives A000031.
Sequence in context: A298637 A034867 A323893 * A193790 A055446 A104706
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Nov 30 2019
STATUS
approved