A089254 -id:A089254 - 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

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

Original entry on oeis.org

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.

A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 13, 16, 19, 22, 26, 30, 34, 38, 44, 49, 54, 62, 67, 74, 83, 89, 98, 107, 115, 124, 134, 145, 155, 168, 178, 189, 206, 217, 231, 247, 259, 277, 294, 310, 327, 345, 365, 382, 404, 424, 444, 470, 489, 513, 539, 561, 588, 613, 641, 670, 699, 729, 756, 791, 824

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Number of partitions of n into distinct terms of A000009, where 2 different parts of 1 and 2 different parts of 2 are available (1a, 1b, 2a, 2b, 3a, 4a, 5a, 6a, ...).

			a(3) = 5 because we have [3a], [2a, 1a], [2a, 1b], [2b, 1a] and [2b, 1b].

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for related partition-counting sequences

Cf. A000009, A007279, A050342, A068006, A089254, A089259, A280253, A284908.

N:= 20: # to get a(0) .. a(A000009(N))
P:= mul(1+x^k,k=1..N):
R:= mul(1+x^coeff(P,x,n)),n=1..N):
seq(coeff(R,x,n),n=0..coeff(P,x,N)); # Robert Israel, Sep 01 2017

nmax = 61; CoefficientList[Series[Product[1 + x^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]

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

A304783 Expansion of Product_{k>=1} (1 - x^k)^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, -1, -1, -1, 0, 1, 0, 3, 2, 3, 1, 3, -2, 0, -6, -8, -12, -14, -18, -19, -19, -15, -3, 4, 29, 46, 90, 114, 165, 192, 248, 252, 276, 232, 185, 29, -143, -454, -811, -1324, -1909, -2609, -3348, -4132, -4851, -5386, -5653, -5380, -4470, -2477, 664, 5582, 12193, 21314

Offset: 0

Ilya Gutkovskiy, May 18 2018

sign

Convolution inverse of A089259.

Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for sequences related to partitions

Cf. A000009, A050342, A089254, A089259, A300508.

nmax = 53; CoefficientList[Series[Product[(1 - x^k)^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 53}]

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

A304784 Expansion of Product_{k>=1} 1/(1 + x^k)^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, -1, -1, -2, 0, -1, 2, 3, 11, 8, 19, 13, 22, -5, -10, -80, -105, -246, -303, -502, -506, -681, -400, -231, 873, 1956, 4733, 7536, 12891, 17609, 25188, 29508, 34890, 29690, 19039, -17742, -74002, -183563, -333665, -572271, -866683, -1271429, -1698491, -2181207

Offset: 0

Ilya Gutkovskiy, May 18 2018

sign

Convolution inverse of A261049.

Eric Weisstein's World of Mathematics, Partition Function P
Index entries for sequences related to partitions

Cf. A000041, A001970, A089254, A261049, A300508.

nmax = 43; CoefficientList[Series[Product[1/(1 + x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d PartitionsP[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

G.f.: Product_{k>=1} 1/(1 + x^k)^A000041(k).

A316231 Expansion of Product_{k>=1} 1/(1 + q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, -1, 0, -2, 1, -2, 3, -3, 6, -8, 14, -10, 28, -26, 41, -73, 90, -112, 155, -221, 288, -501, 560, -799, 1153, -1610, 1953, -3095, 4073, -5224, 7295, -9536, 13536, -18402, 24757, -32936, 48714, -60790, 82101, -113247, 153330, -201522, 275713, -367041, 492991

Offset: 0

Ilya Gutkovskiy, Jun 27 2018

sign

Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for sequences related to partitions

Cf. A000009, A089254, A270995, A279785, A304786, A316230.

nmax = 44; CoefficientList[Series[Product[1/(1 + PartitionsQ[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[Sum[(-1)^k PartitionsQ[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-PartitionsQ[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 44}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*q(j)^k*x^(j*k)/k).

cp's OEIS Frontend

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

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

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A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

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A304783 Expansion of Product_{k>=1} (1 - x^k)^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

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A304784 Expansion of Product_{k>=1} 1/(1 + x^k)^p(k), where p(k) = number of partitions of k (A000041).

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A316231 Expansion of Product_{k>=1} 1/(1 + q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

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