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.

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.

Links

Table of n, a(n) for n=0..77.

Kent E. Morrison, Random polynomials over finite fields, 1999.

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

Cf. A001037, A356583.

Sequence in context: A107030 A371692 A271362 * A263643 A336534 A271852

Adjacent sequences: A354552 A354553 A354554 * A354556 A354557 A354558

Keyword

nonn,tabl

Author

Geoffrey Critzer, Aug 18 2022

Status

approved