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A306945 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 squarefree factors in its unique factorization into irreducible polynomials. 3
2, 1, 1, 2, 2, 3, 4, 1, 6, 8, 2, 9, 16, 7, 18, 30, 14, 2, 30, 60, 34, 4, 56, 114, 72, 14, 99, 220, 156, 36, 1, 186, 422, 320, 90, 6, 335, 817, 671, 207, 18, 630, 1564, 1364, 484, 54, 1161, 3023, 2787, 1070, 148, 3, 2182, 5818, 5624, 2362, 386, 12, 4080, 11240, 11357, 5095, 947, 49 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

T(n,k) is also the number of binary words of length n whose Lyndon factorization is strict, i.e., it contains exactly k factors of distinct Lyndon words.

LINKS

Alois P. Heinz, Rows n = 1..500, flattened

FORMULA

G.f.: Product_{k>=1} (1 + y*x)^A001037(k).

EXAMPLE

Triangular array T(n,k) begins:

   2;

   1,   1;

   2,   2;

   3,   4,   1;

   6,   8,   2;

   9,  16,   7;

  18,  30,  14,  2;

  30,  60,  34,  4;

  56, 114,  72, 14;

  99, 220, 156, 36, 1;

  ...

MAPLE

with(numtheory):

g:= proc(n) option remember; `if`(n=0, 1,

      add(mobius(n/d)*2^d, d=divisors(n))/n)

    end:

b:= proc(n, i) option remember; expand(`if`(n=0, x^n, `if`(i<1, 0,

      add(binomial(g(i), j)*b(n-i*j, i-1)*x^j, j=0..n/i))))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):

seq(T(n), n=1..20);  # Alois P. Heinz, May 28 2019

MATHEMATICA

nn = 16; a = Table[1/n Sum[2^d MoebiusMu[n/d], {d, Divisors[n]}], {n, 1, nn}]; Map[Select[#, # > 0 &] &, Drop[CoefficientList[

    Series[Product[ (1 + u z^k)^a[[k]], {k, 1, nn}], {z, 0, nn}], {z, u}], 1]] // Grid

CROSSREFS

Column k=1 gives A001037.

Row sums give A090129(n+1).

Cf. A269456.

Sequence in context: A231561 A113297 A119985 * A234716 A181885 A116560

Adjacent sequences:  A306942 A306943 A306944 * A306946 A306947 A306948

KEYWORD

nonn,look,tabf

AUTHOR

Geoffrey Critzer, Mar 25 2019

STATUS

approved

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Last modified January 25 23:08 EST 2020. Contains 331270 sequences. (Running on oeis4.)