A285229 -id:A285229 - cp's OEIS frontend

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

1, 1, 2, 4, 7, 12, 22, 36, 61, 101, 166, 267, 433, 686, 1088, 1709, 2671, 4140, 6403, 9824, 15028, 22864, 34657, 52288, 78646, 117784, 175865, 261657, 388145, 573936, 846377, 1244475, 1825170, 2669776, 3895833, 5671127, 8236945, 11936594, 17261557, 24909756

Offset: 0

N. J. A. Sloane, Dec 23 2003

nonn

Number of complete set partitions of the integer partitions of n. This is the Euler transform of A000009. If we change the combstruct command from unlabeled to labeled, then we get A000258. - Thomas Wieder, Aug 01 2008

Number of set multipartitions (multisets of sets) of integer partitions of n. Also a(n) < A270995(n) for n>5. - Gus Wiseman, Apr 10 2016

			From _Gus Wiseman_, Oct 22 2018: (Start)
The a(6) = 22 set multipartitions of integer partitions of 6:
  (6)  (15)    (123)      (12)(12)      (1)(1)(1)(12)    (1)(1)(1)(1)(1)(1)
       (24)    (1)(14)    (1)(1)(13)    (1)(1)(1)(1)(2)
       (1)(5)  (1)(23)    (1)(2)(12)
       (2)(4)  (2)(13)    (1)(1)(1)(3)
       (3)(3)  (3)(12)    (1)(1)(2)(2)
               (1)(1)(4)
               (1)(2)(3)
               (2)(2)(2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row sums of A285229 and of A360763.

Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.

with(combstruct): A089259:= [H, {H=Set(T, card>=1), T=PowerSet (Sequence (Z, card>=1), card>=1)}, unlabeled]; 1, seq (count (A089259, size=j), j=1..16); # Thomas Wieder, Aug 01 2008
# second Maple program:
with(numtheory):
b:= proc(n, i)
      if n<0 or n>i*(i+1)/2 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n,i):= b(n-i, i-1) +b(n, i-1)
      fi
    end:
a:= proc(n) option remember; `if` (n=0, 1,
       add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 11 2011

max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 24 2014 *)
b[n_, i_] := b[n, i] = Which[n<0 || n>i*(i+1)/2, 0, n == 0, 1, i<1, 0, True, b[n-i, i-1] + b[n, i-1]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d* b[d, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat([1], EulerT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)))} \\ Andrew Howroyd, Oct 26 2018

A061260 G.f.: Product_{k>=1} (1-y*x^k)^(-numbpart(k)), where numbpart(k) = number of partitions of k, cf. A000041.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 6, 2, 1, 7, 11, 6, 2, 1, 11, 23, 15, 6, 2, 1, 15, 40, 32, 15, 6, 2, 1, 22, 73, 67, 37, 15, 6, 2, 1, 30, 120, 134, 79, 37, 15, 6, 2, 1, 42, 202, 255, 172, 85, 37, 15, 6, 2, 1, 56, 320, 470, 348, 187, 85, 37, 15, 6, 2, 1, 77, 511, 848, 697, 397, 194, 85, 37, 15, 6, 2, 1

Offset: 1

Vladeta Jovovic, Apr 23 2001

Multiset transformation of A000041. - R. J. Mathar, Apr 30 2017

Number of orderless twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. The T(5,3) = 6 orderless twice-partitions: (3)(1)(1), (21)(1)(1), (111)(1)(1), (2)(2)(1), (2)(11)(1), (11)(11)(1). - Gus Wiseman, Mar 23 2018

			:  1;
:  2,   1;
:  3,   2,   1;
:  5,   6,   2,   1;
:  7,  11,   6,   2,  1;
: 11,  23,  15,   6,  2,  1;
: 15,  40,  32,  15,  6,  2,  1;
: 22,  73,  67,  37, 15,  6,  2, 1;
: 30, 120, 134,  79, 37, 15,  6, 2, 1;
: 42, 202, 255, 172, 85, 37, 15, 6, 2, 1;

Alois P. Heinz, Rows n = 1..141, flattened
Index entries for triangles generated by the Multiset Transformation

Row sums: A001970, first column: A000041.
T(2,n) gives A061261,
Cf. A063834, A119442, A273873, A285229, A289078, A289501, A299200, A299201.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*binomial(
       combinat[numbpart](i)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[PartitionsP[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 17 2018, after Alois P. Heinz *)

A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 0, 7, 11, 1, 0, 0, 0, 11, 20, 6, 0, 0, 0, 0, 15, 40, 16, 0, 0, 0, 0, 0, 22, 68, 40, 3, 0, 0, 0, 0, 0, 30, 120, 91, 11, 0, 0, 0, 0, 0, 0, 42, 195, 186, 41, 0, 0, 0, 0, 0, 0, 0, 56, 320, 367, 105, 3, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
  1
  0  1
  0  2  0
  0  3  2  0
  0  5  4  0  0
  0  7 11  1  0  0
  0 11 20  6  0  0  0
  0 15 40 16  0  0  0  0
  0 22 68 40  3  0  0  0  0
  ...
Row n = 5 counts the following sets of multisets:
  {{5}}          {{1},{4}}        {{1},{2},{1,1}}
  {{1,4}}        {{2},{3}}
  {{2,3}}        {{1},{1,3}}
  {{1,1,3}}      {{1},{2,2}}
  {{1,2,2}}      {{2},{1,2}}
  {{1,1,1,2}}    {{3},{1,1}}
  {{1,1,1,1,1}}  {{1},{1,1,2}}
                 {{1,1},{1,2}}
                 {{2},{1,1,1}}
                 {{1},{1,1,1,1}}
                 {{1,1},{1,1,1}}

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

Row sums are A261049.
Column k = 1 is A000041.
Multisets of multisets are A061260, with row sums A001970.
Sets of sets are A330462, with row sums A050342.
Multisets of sets are A285229, with row sums A089259.
Sets of disjoint sets are A330460, with row sums A294617.
Cf. A007716, A060016, A063834, A255906, A271619, A279785, A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
       combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 30 2019

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@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
     PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(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)^A000041(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.

A285230 Number of multisets of exactly n partitions of positive integers into distinct parts with total sum of parts equal to 2n.

Original entry on oeis.org

1, 1, 3, 5, 11, 19, 37, 63, 115, 195, 339, 566, 957, 1573, 2599, 4217, 6842, 10962, 17531, 27767, 43862, 68769, 107469, 166942, 258461, 398124, 611237, 934356, 1423724, 2161145, 3270560, 4932647, 7418099, 11121610, 16629101, 24794130, 36874451, 54698714

Offset: 0

Alois P. Heinz, Apr 14 2017

nonn

			a(3) = 5: {4,1,1}, {31,1,1}, {3,2,1}, {21,2,1}, {2,2,2}.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009, A285229.

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*g(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d + 1], {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

G.f.: Product_{j>=1} 1/(1-x^j)^A000009(j+1).

a(n) = A285229(2n,n).

A320787 Number of multisets of exactly two partitions of positive integers into distinct parts with total sum of parts equal to n.

Original entry on oeis.org

1, 1, 3, 4, 8, 11, 18, 25, 38, 52, 75, 101, 140, 186, 252, 330, 438, 567, 740, 948, 1221, 1549, 1973, 2482, 3129, 3907, 4884, 6055, 7512, 9255, 11402, 13967, 17102, 20836, 25372, 30760, 37262, 44970, 54221, 65156, 78220, 93622, 111937, 133481, 158996, 188930

Offset: 2

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A285229.

Cf. A000009.

g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
      d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 3)
    end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=2..60);

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*x^j*Binomial[g[i] + j - 1, j], {j, 0, n/i}]]], {x, 0, 3}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 2}];
a /@ Range[2, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) = [x^n y^2] Product_{j>=1} 1/(1-y*x^j)^A000009(j).

A320788 Number of multisets of exactly three partitions of positive integers into distinct parts with total sum of parts equal to n.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 29, 44, 72, 110, 169, 250, 373, 538, 778, 1104, 1559, 2172, 3016, 4136, 5651, 7653, 10314, 13800, 18389, 24342, 32097, 42096, 54991, 71500, 92637, 119506, 153659, 196831, 251332, 319834, 405824, 513312, 647504, 814448, 1021792, 1278547

Offset: 3

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A285229.

Cf. A000009.

g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
      d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 4)
    end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=3..60);

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*x^j*Binomial[g[i] + j - 1, j], {j, 0, n/i}]]], {x, 0, 4}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 3}];
a /@ Range[3, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) = [x^n y^3] Product_{j>=1} 1/(1-y*x^j)^A000009(j).

A320789 Number of multisets of exactly four partitions of positive integers into distinct parts with total sum of parts equal to n.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 34, 55, 96, 152, 248, 386, 607, 921, 1405, 2092, 3112, 4551, 6635, 9545, 13683, 19401, 27393, 38346, 53441, 73928, 101840, 139398, 190020, 257601, 347836, 467381, 625686, 833917, 1107547, 1465136, 1931754, 2537747, 3323490, 4338012, 5645645

Offset: 4

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A285229.

Cf. A000009.

g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
      d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 5)
    end:
a:= n-> coeff(b(n$2), x, 4):
seq(a(n), n=4..60);

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*x^j*Binomial[g[i] + j - 1, j], {j, 0, n/i}]]], {x, 0, 5}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 4}];
a /@ Range[4, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) = [x^n y^4] Product_{j>=1} 1/(1-y*x^j)^A000009(j).

A320790 Number of multisets of exactly five partitions of positive integers into distinct parts with total sum of parts equal to n.

Original entry on oeis.org

1, 1, 3, 5, 11, 19, 36, 60, 107, 176, 296, 475, 770, 1211, 1906, 2939, 4518, 6842, 10313, 15363, 22770, 33424, 48802, 70688, 101854, 145755, 207528, 293704, 413691, 579571, 808328, 1121923, 1550645, 2133751, 2924579, 3992307, 5429751, 7357195, 9934357

Offset: 5

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A285229.

Cf. A000009.

g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
      d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 6)
    end:
a:= n-> coeff(b(n$2), x, 5):
seq(a(n), n=5..60);

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*x^j*Binomial[g[i] + j - 1, j], {j, 0, n/i}]]], {x, 0, 6}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 5}];
a /@ Range[5, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) = [x^n y^5] Product_{j>=1} 1/(1-y*x^j)^A000009(j).

A320791 Number of multisets of exactly six partitions of positive integers into distinct parts with total sum of parts equal to n.

Original entry on oeis.org

1, 1, 3, 5, 11, 19, 37, 62, 112, 187, 320, 523, 866, 1386, 2229, 3510, 5516, 8538, 13172, 20073, 30461, 45781, 68469, 101586, 149991, 219922, 320925, 465492, 672055, 965063, 1379741, 1962957, 2781094, 3922672, 5511041, 7710818, 10748577, 14926037, 20654385

Offset: 6

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A285229.

Cf. A000009.

g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
      d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 7)
    end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=6..60);

a(n) = [x^n y^6] Product_{j>=1} 1/(1-y*x^j)^A000009(j).

cp's OEIS Frontend

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

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A061260 G.f.: Product_{k>=1} (1-y*x^k)^(-numbpart(k)), where numbpart(k) = number of partitions of k, cf. A000041.

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A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets 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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A285230 Number of multisets of exactly n partitions of positive integers into distinct parts with total sum of parts equal to 2n.

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A320787 Number of multisets of exactly two partitions of positive integers into distinct parts with total sum of parts equal to n.

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A320788 Number of multisets of exactly three partitions of positive integers into distinct parts with total sum of parts equal to n.

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A320789 Number of multisets of exactly four partitions of positive integers into distinct parts with total sum of parts equal to n.