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A354555
Rectangular array read by antidiagonals. T(m,n) is the number of degree n monic polynomials in GF_2[x] such that each irreducible factor in the prime factorization has multiplicity no greater than m, m>=1, n>=0.
0
1, 1, 2, 1, 2, 2, 1, 2, 4, 4, 1, 2, 4, 6, 8, 1, 2, 4, 8, 12, 16, 1, 2, 4, 8, 14, 24, 32, 1, 2, 4, 8, 16, 28, 48, 64, 1, 2, 4, 8, 16, 30, 56, 96, 128, 1, 2, 4, 8, 16, 32, 60, 112, 192, 256, 1, 2, 4, 8, 16, 32, 62, 120, 224, 384, 512, 1, 2, 4, 8, 16, 32, 64, 124, 240, 448, 768, 1024
OFFSET
0,3
COMMENTS
Row m = 1 counts the squarefree monic polynomials of degree n in GF_2[x] which is the main diagonal of A356583.
FORMULA
For n>m, T(m,n) = 2^n - 2^(n-m).
G.f. for row m: (1/(1-2x))*Product_{n>=1}(1-x^(n(m+1)))^A001037(n).
EXAMPLE
1, 2, 2, 4, 8, 16, 32, 64, 128, 256, 512,
1, 2, 4, 6, 12, 24, 48, 96, 192, 384, 768,
1, 2, 4, 8, 14, 28, 56, 112, 224, 448, 896,
1, 2, 4, 8, 16, 30, 60, 120, 240, 480, 960,
1, 2, 4, 8, 16, 32, 62, 124, 248, 496, 992,
1, 2, 4, 8, 16, 32, 64, 126, 252, 504, 1008,
1, 2, 4, 8, 16, 32, 64, 128, 254, 508, 1016,
1, 2, 4, 8, 16, 32, 64, 128, 256, 510, 1020,
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1022,
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
MATHEMATICA
nn = 12; a[q_, r_] := 1/r Sum[MoebiusMu[r/d] q^d, {d, Divisors[r]}]; mfree =
Table[CoefficientList[Series[1/(1 -q t) Product[((1 - t^n) Sum[(t^ n)^l, {l, 0, m}])^a[q, n], {n, 1, nn}] /. q -> 2, {t, 0, nn}], t], {m, 1, nn}];
Table[Table[mfree[[m + 1 - i, i]], {i, 1, m}], {m, 1, nn}] // Flatten
CROSSREFS
Sequence in context: A107030 A371692 A271362 * A263643 A336534 A271852
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Aug 18 2022
STATUS
approved