A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 4, 1, 6, 1, 1, 1, 1, 2, 6, 1, 5, 1, 7, 1, 1, 1, 1, 2, 3, 10, 1, 6, 1, 8, 1, 1, 1, 1, 1, 3, 4, 15, 1, 7, 1, 9, 1, 1, 1, 1, 4, 1, 4, 5, 21, 1, 8, 1, 10, 1, 1, 1

Offset

1,12

Comments

An orderless factorization of n with k > 1 levels of parentheses is any multiset partition of an orderless factorization of n with k - 1 levels of parentheses. If k = 1 it is just an orderless factorization of n into factors > 1.

Links

Table of n, a(n) for n=1..91.

Example

Array begins:
       k=0  k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9  k=10 k=11 k=12
   n=1: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=2: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=3: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=4: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=5: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=6: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=7: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=8: 1    3    6   10   15   21   28   36   45   55   66   78   91
   n=9: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=10: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=11: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=12: 1    4    9   16   25   36   49   64   81  100  121  144  169
  n=13: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=14: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=15: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=16: 1    5   14   30   55   91  140  204  285  385  506  650  819
  n=17: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=18: 1    4    9   16   25   36   49   64   81  100  121  144  169
The A(12,3) = 16 orderless factorizations of 12 with 2 levels of parentheses:
  ((2*2*3))          ((2*6))      ((3*4))      ((12))
  ((2)*(2*3))        ((2)*(6))    ((3)*(4))
  ((3)*(2*2))        ((2))*((6))  ((3))*((4))
  ((2))*((2*3))
  ((2)*(2)*(3))
  ((3))*((2*2))
  ((2))*((2)*(3))
  ((3))*((2)*(2))
  ((2))*((2))*((3))

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
lev[n_, k_]:=If[k==0, {n}, Join@@Table[Union[Sort/@Tuples[lev[#, k-1]&/@fac]], {fac, facs[n]}]];
Table[Length[lev[sum-k, k]], {sum, 12}, {k, 0, sum-1}]

Crossrefs

Columns: A001055 (k=1), A050336 (k=2), A050338 (k=3), A050340 (k=4).

Rows: A000027, A000217, A000290, etc.

Cf. A096751, A141268, A144150, A213427, A255906, A281113, A290353, A292504, A317145, A318564, A318565, A318812, A323718.

Sequence in context: A033630 A308608 A371213 * A220122 A101446 A333769

Adjacent sequences: A323716 A323717 A323718 * A323720 A323721 A323722

Keyword

nonn,tabl

Author

Gus Wiseman, Jan 25 2019

Status

approved