A323718 -id:A323718 - cp's OEIS frontend

A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.

1, 1, 3, 6, 15, 28, 66, 122, 266, 503, 1027, 1913, 3874, 7099, 13799, 25501, 48508, 88295, 165942, 299649, 554545, 997281, 1817984, 3245430, 5875438, 10410768, 18635587, 32885735, 58399350, 102381103, 180634057, 314957425, 551857780, 958031826, 1667918758

Offset: 0

Wouter Meeussen, Aug 21 2001

These are different from plane partitions.
For ordered partitions of partitions see A055887 which may be computed from A036036 and A048996. - Alford Arnold, May 19 2006
Twice partitioned numbers correspond to triangles (or compositions) in the multiorder of integer partitions. - Gus Wiseman, Oct 28 2015

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + 266*x^8 + ...
If n=6, a possible first partitioning is (3+3), resulting in the following second partitionings: ((3),(3)), ((3),(2+1)), ((3),(1+1+1)), ((2+1),(3)), ((2+1),(2+1)), ((2+1),(1+1+1)), ((1+1+1),(3)), ((1+1+1),(2+1)), ((1+1+1),(1+1+1)).

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n=1..500 from T. D. Noe)
Vaclav Kotesovec, Screenshot - A closed form formula for the constant c
Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Sequences enumerating triangles of integer partitions
Gus Wiseman, On the Categorical Structure of Weakly Ordered Sequences of Integer Partitions

Cf. A063835, A196545.
Cf. A036036, A048996, A055887.
Cf. A006906, A270995.
Cf. A007425, A047966, A047968, A271619, A279375, A279784-A279791.
The strict case is A296122.
Row sums of A321449.
Column k=2 of A323718.
Without singletons we have A327769, A358828, A358829.
For odd lengths we have A358823, A358824.
For distinct lengths we have A358830, A358912.
For strict partitions see A358914, A382524.
A000041 counts integer partitions, strict A000009.
A001970 counts multiset partitions of integer partitions.
Cf. A075900, A358831, A358833, A358906, A358908, A358909.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+`if`(i>n, 0, numbpart(i)*b(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 26 2015

Table[Plus @@ Apply[Times, IntegerPartitions[i] /. i_Integer :> PartitionsP[i], 2], {i, 36}]
(* second program: *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i > n, 0, PartitionsP[i]*b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 19 2016 */

G.f.: 1/Product_{k>0} (1-A000041(k)*x^k). n*a(n) = Sum_{k=1..n} b(k)*a(n-k), a(0) = 1, where b(k) = Sum_{d|k} d*A000041(d)^(k/d) = 1, 5, 10, 29, 36, 110, 106, ... . - Vladeta Jovovic, Jun 19 2003
From Vaclav Kotesovec, Mar 27 2016: (Start)
a(n) ~ c * 5^(n/4), where
c = 96146522937.7161898848278970039269600938032826... if n mod 4 = 0
c = 96146521894.9433858914667933636782092683849082... if n mod 4 = 1
c = 96146522937.2138934755566928890704687838407524... if n mod 4 = 2
c = 96146521894.8218716328341714149619262713426755... if n mod 4 = 3
(End)

a(0)=1 prepended by Alois P. Heinz, Nov 26 2015

A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 33, 21, 8, 1, 11, 91, 104, 36, 10, 1, 15, 298, 452, 238, 55, 12, 1, 22, 910, 2335, 1430, 455, 78, 14, 1, 30, 3017, 11992, 10179, 3505, 775, 105, 16, 1, 42, 9945, 66810, 74299, 31881, 7297, 1218, 136, 18, 1, 56

Offset: 1

Gus Wiseman, Jan 27 2019

			Array begins:
      k=1:  k=2:  k=3:  k=4:  k=5:  k=6:
  n=1:  1     1     1     1     1     1
  n=2:  2     4     6     8    10    12
  n=3:  3    10    21    36    55    78
  n=4:  5    33   104   238   455   775
  n=5:  7    91   452  1430  3505  7297
  n=6: 11   298  2335 10179 31881 80897
Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:
  {{111}}          {{112}}          {{123}}
  {{1}{11}}        {{1}{12}}        {{1}{23}}
  {{1}}{{11}}      {{2}{11}}        {{1}}{{23}}
  {{1}{1}{1}}      {{1}}{{12}}      {{1}{2}{3}}
  {{1}}{{1}{1}}    {{1}{1}{2}}      {{1}}{{2}{3}}
  {{1}}{{1}}{{1}}  {{2}}{{11}}      {{1}}{{2}}{{3}}
                   {{1}}{{1}{2}}
                   {{2}}{{1}{1}}
                   {{1}}{{1}}{{2}}

Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).
Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).
Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Apply[Rule,Table[{undats[m][[i]],i},{i,Length[undats[m]]}],{1}]],First[Sort[expnorm[m,1]]]]];
expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#1>=aft&]}]},Union@@(expnorm[#1,aft+1]&)/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,[__]]],{par,First/@Position[mx,Max[mx]]}]]]];
strnorm[n_]:=(Flatten[MapIndexed[Table[#2,{#1}]&,#1]]&)/@IntegerPartitions[n];
kmp[n_,k_]:=kmp[n,k]=If[k==1,strnorm[n],Union[expnorm/@Join@@mps/@kmp[n,k-1]]];
Table[Length[kmp[sum-k,k]],{sum,1,7},{k,1,sum-1}]

a(46)-a(56) from Robert Price, May 11 2021

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1

Offset: 0

Paul D. Hanna, Jul 07 2004

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

			n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Pontus von Brömssen, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).
Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
Cf. A096651, A096752, A096753.
Cf. A096806.
Cf. A042984.
Cf. A144150, A213427, A290353, A323718, A323719.

trans[x_]:=If[x=={},{},Transpose[x]];
levptns[n_,k_]:=If[k==1,IntegerPartitions[n],Join@@Table[Select[Tuples[levptns[#,k-1]&/@y],And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#,ConstantArray[n,k-1]]&/@#)])&],{y,IntegerPartitions[n]}]];
Table[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* Gus Wiseman, Jan 27 2019 *)

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.

Inverse binomial transforms of the columns is given by triangle A096806.

A327639 Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 15, 16, 6, 1, 10, 45, 88, 76, 24, 1, 14, 93, 282, 420, 302, 84, 1, 21, 223, 1052, 2489, 3112, 1970, 498, 1, 29, 444, 2950, 9865, 18123, 18618, 10046, 2220, 1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108

Offset: 0

Alois P. Heinz, Sep 20 2019

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
Row n is the inverse binomial transform of the n-th row of array A323718.

			T(4,0) = 1:  4
T(4,1) = 4:     T(4,2) = 6:          T(4,3) = 3:
  4-> 31          4-> 31 -> 211        4-> 31 -> 211 -> 1111
  4-> 22          4-> 31 -> 1111       4-> 22 -> 112 -> 1111
  4-> 211         4-> 22 -> 112        4-> 22 -> 211 -> 1111
  4-> 1111        4-> 22 -> 211
                  4-> 22 -> 1111
                  4-> 211-> 1111
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2,   1;
  1,  4,   6,    3;
  1,  6,  15,   16,     6;
  1, 10,  45,   88,    76,     24;
  1, 14,  93,  282,   420,    302,     84;
  1, 21, 223, 1052,  2489,   3112,   1970,    498;
  1, 29, 444, 2950,  9865,  18123,  18618,  10046,  2220;
  1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108;
  ...

Alois P. Heinz, Rows n = 0..170, flattened
Wikipedia, Iverson bracket
Wikipedia, Partition (number theory)

Columns k=0-2 give A000012, A000065, A327769.
Row sums give A327644.
T(2n,n) gives A327645.
Cf. A323718, A327631, A327643, A327646.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, Max[0, n - 1] }] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A323718(n,i).
T(n,n-1) = A327631(n,n-1)/n = A327643(n) for n >= 1.
Sum_{k=1..n-1} k * T(n,k) = A327646(n).
Sum_{k=0..max(0,n-1)} (-1)^k * T(n,k) = [n<2], where [] is an Iverson bracket.

A327618 Number A(n,k) of parts in all k-times partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 14, 12, 1, 0, 1, 9, 25, 44, 20, 1, 0, 1, 11, 39, 109, 100, 35, 1, 0, 1, 13, 56, 219, 315, 274, 54, 1, 0, 1, 15, 76, 386, 769, 1179, 581, 86, 1, 0, 1, 17, 99, 622, 1596, 3643, 3234, 1417, 128, 1, 0, 1, 19, 125, 939, 2960, 9135, 12336, 10789, 2978, 192, 1

Offset: 0

Alois P. Heinz, Sep 19 2019

Row n is binomial transform of the n-th row of triangle A327631.

			A(2,2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
Square array A(n,k) begins:
  0,  0,   0,    0,     0,     0,     0,      0, ...
  1,  1,   1,    1,     1,     1,     1,      1, ...
  1,  3,   5,    7,     9,    11,    13,     15, ...
  1,  6,  14,   25,    39,    56,    76,     99, ...
  1, 12,  44,  109,   219,   386,   622,    939, ...
  1, 20, 100,  315,   769,  1596,  2960,   5055, ...
  1, 35, 274, 1179,  3643,  9135, 19844,  38823, ...
  1, 54, 581, 3234, 12336, 36911, 93302, 208377, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-3 give: A057427, A006128, A327594, A327627.
Rows n=0-3 give: A000004, A000012, A005408, A095794(k+1).
Main diagonal gives A327619.
Cf. A323718, A327622, A327631.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
         (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
    end:
A:= (n, k)-> b(n$2, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/ h[[1]]}][h[[1]] b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
A[n_, k_] := b[n, n, k][[2]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327631(n,i).

A301595 Number of thrice-partitions of n.

Original entry on oeis.org

1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508

Offset: 0

Gus Wiseman, Mar 24 2018

nonn

A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.

			The a(3) = 10 thrice-partitions:
  ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
  ((2))((1)), ((11))((1)), ((1)(1))((1)),
  ((1))((1))((1)).

Alois P. Heinz, Table of n, a(n) for n = 0..3244
Gus Wiseman, The a(4) = 34 thrice-partitions of 4.

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.

Column k=3 of A323718.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 25 2019

twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
Array[thrie,30]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
     1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
a[n_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

A306187 Number of n-times partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 65, 371, 3780, 33552, 472971, 5736082, 97047819, 1547576394, 32992294296, 626527881617, 15202246707840, 352290010708120, 9970739854456849, 262225912049078193, 8309425491887714632, 250946978120046026219, 8898019305511325083149

Offset: 0

Alois P. Heinz, Jan 27 2019

nonn

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n. The only 0-times partition of n is the number n itself. - Gus Wiseman, Jan 27 2019

			From _Gus Wiseman_, Jan 27 2019: (Start)
The a(1) = 1 through a(3) = 10 partitions:
  (1)  ((2))     (((3)))
       ((11))    (((21)))
       ((1)(1))  (((111)))
                 (((2)(1)))
                 (((11)(1)))
                 (((2))((1)))
                 (((1)(1)(1)))
                 (((11))((1)))
                 (((1)(1))((1)))
                 (((1))((1))((1)))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..410
Wikipedia, Partition (number theory)

Main diagonal of A323718.
Cf. A000041, A306188.
Cf. A001970, A063834, A096752, A196545, A261280, A289501, A290354, A327619.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);

ptnlevct[n_,k_]:=Switch[k,0,1,1,PartitionsP[n],_,SeriesCoefficient[Product[1/(1-ptnlevct[m,k-1]*x^m),{m,n}],{x,0,n}]];
Table[ptnlevct[n,n],{n,0,8}] (* Gus Wiseman, Jan 27 2019 *)

a(n) = A323718(n,n).

A330461 Array read by antidiagonals where A(n,k) is the number of multiset partitions with k levels that are strict at all levels and have total sum n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 4, 7, 7, 5, 1, 1, 1, 1, 5, 12, 14, 11, 6, 1, 1, 1, 1, 6, 19, 29, 25, 16, 7, 1, 1, 1, 1, 8, 30, 57, 60, 41, 22, 8, 1, 1, 1, 1, 10, 49, 110, 141, 111, 63, 29, 9, 1, 1, 1

Offset: 0

Gus Wiseman, Dec 18 2019

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6
      -----------------------------
  n=0:  1   1   1   1   1   1   1
  n=1:  1   1   1   1   1   1   1
  n=2:  1   1   1   1   1   1   1
  n=3:  1   2   3   4   5   6   7
  n=4:  1   2   4   7  11  16  22
  n=5:  1   3   7  14  25  41  63
  n=6:  1   4  12  29  60 111 189
For example, the A(5,3) = 14 partitions are:
  {{5}}      {{1}}{{4}}
  {{14}}     {{2}}{{3}}
  {{23}}     {{1}}{{13}}
  {{1}{4}}   {{2}}{{12}}
  {{2}{3}}   {{1}}{{1}{3}}
  {{1}{13}}  {{2}}{{1}{2}}
  {{2}{12}}  {{1}}{{1}{12}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns are A000012 (k = 0), A000009 (k = 1), A050342 (k = 2), A050343 (k = 3), A050344 (k = 4).
The non-strict version is A290353.
Cf. A001970, A004111, A007713, A060016, A273873, A279375, A279785, A294617, A306186, A323718, A323790, A330462.

spl[n_,0]:={n};
spl[n_,k_]:=Select[Join@@Table[Union[Sort/@Tuples[spl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}],UnsameQ@@#&];
Table[Length[spl[n-k,k]],{n,0,10},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
M(n, k=n)={my(L=List(), v=vector(n,i,1)); listput(L, concat([1], v)); for(j=1, k, v=WeighT(v); listput(L, concat([1], v))); Mat(Col(L))~}
{ my(A=M(7)); for(i=1, #A, print(A[i,])) } \\ Andrew Howroyd, Dec 31 2019

Column k is the k-th weigh transform of the all-ones sequence. The weigh transform of a sequence b has generating function Product_{i > 0} (1 + x^i)^b(i).

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

Original entry on oeis.org

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

Gus Wiseman, Jan 25 2019

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.

			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))

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.

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}]

cp's OEIS Frontend

A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.

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A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

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A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

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A327639 Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.

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A327618 Number A(n,k) of parts in all k-times partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A301595 Number of thrice-partitions of n.

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A306187 Number of n-times partitions of n.

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