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A251683 Irregular triangular array: T(n,k) is the number of ordered factorizations of n with exactly k factors, n >= 1, 1 <= k <= A086436(n). 7
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 2, 1, 2, 1, 3, 3, 1, 1, 1, 4, 3, 1, 1, 4, 3, 1, 2, 1, 2, 1, 1, 6, 9, 4, 1, 1, 1, 2, 1, 2, 1, 1, 4, 3, 1, 1, 6, 6, 1, 1, 4, 6, 4, 1, 1, 2, 1, 2, 1, 2, 1, 7, 12, 6, 1, 1, 2, 1, 2, 1, 6, 9, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Row sums = A074206.

Row lengths give A086436.

T(n,2) = A070824(n).

T(n,3) = A200221(n).

Sum_{k>=1} k*T(n,k) = A254577.

For all n > 1,  Sum_{k=1..A086436(n)} (-1)^k*T(n,k) = A008683(n). - Geoffrey Critzer, May 25 2018

LINKS

Alois P. Heinz, Rows n = 1..4000, flattened

Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).

Eric Weisstein's World of Mathematics, Ordered Factorization

FORMULA

Dirichlet g.f.: 1/(1 - y*(zeta(x)-1)).

EXAMPLE

Triange T(n,k) begins:

  1;

  1;

  1;

  1, 1;

  1;

  1, 2;

  1;

  1, 2, 1;

  1, 1;

  1, 2;

  1;

  1, 4, 3;

  1;

  1, 2;

  1, 2;

  ...

There are 8 ordered factorizations of the integer 12: 12, 6*2, 4*3, 3*4, 2*6, 3*2*2, 2*3*2, 2*2*3.  So T(12,1)=1, T(12,2)=4, and T(12,3)=3.

MAPLE

with(numtheory):

b:= proc(n) option remember; expand(x*(1+

      add(b(n/d), d=divisors(n) minus {1, n})))

    end:

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

seq(T(n), n=1..100);  # Alois P. Heinz, Dec 07 2014

MATHEMATICA

f[1] = {{}};

f[n_] := f[n] =

  Level[Table[

    Map[Prepend[#, d] &, f[n/d]], {d, Rest[Divisors[n]]}], {2}];

Prepend[Map[Select[#, # > 0 &] &,

  Drop[Transpose[

    Table[Map[Count[#, k] &,

      Map[Length, Table[f[n], {n, 1, 40}], {2}]], {k, 1, 10}]],

   1]], {1}] // Grid

CROSSREFS

Cf. A008683, A070824, A074206, A086436, A200221, A254577.

Sequence in context: A276630 A176048 A322480 * A306261 A025430 A256972

Adjacent sequences:  A251680 A251681 A251682 * A251684 A251685 A251686

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Dec 06 2014

STATUS

approved

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Last modified November 19 06:26 EST 2019. Contains 329310 sequences. (Running on oeis4.)