A360764 -id:A360764 - cp's OEIS frontend

A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

1, 1, 1, 3, 4, 7, 12, 19, 30, 49, 77, 119, 186, 286, 438, 670, 1014, 1528, 2300, 3437, 5119, 7603, 11241, 16564, 24343, 35650, 52058, 75820, 110115, 159510, 230522, 332324, 477994, 686044, 982519, 1404243, 2003063, 2851720, 4052429, 5748440, 8140007, 11507125

Offset: 0

Christian G. Bower, Oct 15 1999

nonn

Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.

			4=(4)=(3)+(1)=(3+1)=(2+1)+(1).
From _Gus Wiseman_, Oct 11 2018: (Start)
a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:
  {{1}}  {{2}}  {{3}}      {{4}}        {{5}}        {{6}}
                {{1,2}}    {{1,3}}      {{1,4}}      {{1,5}}
                {{1},{2}}  {{1},{3}}    {{2,3}}      {{2,4}}
                           {{1},{1,2}}  {{1},{4}}    {{1,2,3}}
                                        {{2},{3}}    {{1},{5}}
                                        {{1},{1,3}}  {{2},{4}}
                                        {{2},{1,2}}  {{1},{1,4}}
                                                     {{1},{2,3}}
                                                     {{2},{1,3}}
                                                     {{3},{1,2}}
                                                     {{1},{2},{3}}
                                                     {{1},{2},{1,2}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000
N. J. A. Sloane, Transforms

Cf. A050343-A050350, A089254.
Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.
Row sums of A330462 and of A360764.

g:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)
nn=10;Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k],{k,nn}],{x,0,n}],{n,0,nn}] (* Gus Wiseman, Oct 11 2018 *)

Weigh transform of A000009.

A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 2, 0, 0, 0, 0, 5, 11, 3, 0, 0, 0, 0, 0, 6, 16, 8, 0, 0, 0, 0, 0, 0, 8, 25, 15, 1, 0, 0, 0, 0, 0, 0, 10, 35, 28, 4, 0, 0, 0, 0, 0, 0, 0, 12, 52, 46, 9, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 18 2019

			Triangle begins:
  1
  0  1
  0  1  0
  0  2  1  0
  0  2  2  0  0
  0  3  4  0  0  0
  0  4  6  2  0  0  0
  0  5 11  3  0  0  0  0
  0  6 16  8  0  0  0  0  0
  0  8 25 15  1  0  0  0  0  0
  0 10 35 28  4  0  0  0  0  0  0
  ...
Row n = 7 counts the following set-systems:
  {{7}}      {{1},{6}}      {{1},{2},{4}}
  {{1,6}}    {{2},{5}}      {{1},{2},{1,3}}
  {{2,5}}    {{3},{4}}      {{1},{3},{1,2}}
  {{3,4}}    {{1},{1,5}}
  {{1,2,4}}  {{1},{2,4}}
             {{2},{1,4}}
             {{2},{2,3}}
             {{3},{1,3}}
             {{4},{1,2}}
             {{1},{1,2,3}}
             {{1,2},{1,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

Row sums are A050342.
Column k = 1 is A000009.
Cf. A001970, A050343, A063834, A270995, A271619, A279375, A279785, A283877, A294617, A326031, A330456, A330460, A330463, A360764.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]

L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} (1 + y*x^j)^A000009(j). - Andrew Howroyd, Dec 29 2019

A360763 Number T(n,k) of multisets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 4, 2, 1, 0, 1, 5, 8, 5, 2, 1, 0, 1, 6, 11, 10, 5, 2, 1, 0, 1, 7, 16, 18, 11, 5, 2, 1, 0, 1, 8, 22, 28, 22, 12, 5, 2, 1, 0, 1, 9, 28, 45, 39, 24, 12, 5, 2, 1, 0, 1, 10, 35, 63, 67, 46, 25, 12, 5, 2, 1, 0, 1, 11, 44, 89, 106, 86, 50, 26, 12, 5, 2, 1

Offset: 0

Alois P. Heinz, Feb 19 2023

T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.

Reversed rows and also the columns converge to A360785.

			T(6,1) = 1: {[6]}.
T(6,2) = 5: {[1],[5]}, {[2],[4]}, {[3],[3]}, {[1,5]}, {[2,4]}.
T(6,3) = 8: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[1],[4]}, {[1],[2],[3]}, {[2],[2],[2]}.
T(6,4) = 5: {[1],[1],[1],[3]}, {[1],[1],[2],[2]}, {[1],[1],[1,3]}, {[1],[2],[1,2]}, {[1,2],[1,2]}.
T(6,5) = 2: {[1],[1],[1],[1],[2]}, {[1],[1],[1],[1,2]}.
T(6,6) = 1: {[1],[1],[1],[1],[1],[1]}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 2,  1;
  0, 1, 3,  2,  1;
  0, 1, 4,  4,  2,  1;
  0, 1, 5,  8,  5,  2,  1;
  0, 1, 6, 11, 10,  5,  2,  1;
  0, 1, 7, 16, 18, 11,  5,  2, 1;
  0, 1, 8, 22, 28, 22, 12,  5, 2, 1;
  0, 1, 9, 28, 45, 39, 24, 12, 5, 2, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-2 give: A000007, A057427, A001477(n-1) for n>=1.
Row sums give A089259.
T(2n,n) gives A360784.
T(3n,2n) gives A360785.
Cf. A000009, A008289, A055884, A285229, A360742, A360764.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]];
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i] + k - 1, k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 12 2023, after Alois P. Heinz *)

T(3n,2n) = A360785(n) = T(3n+j,2n+j) for j>=0.

cp's OEIS Frontend

A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

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A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.

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A360763 Number T(n,k) of multisets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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