A270995 -id:A270995 - 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

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

Original entry on oeis.org

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

A381432 Heinz numbers of section-sum partitions. Union of A381431.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A320340, A364347, A350838 in containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}

Partitions of this type are counted by A239455, complement A351293.
The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
Union of A381431 (parts A381436).
The complement is A381433, conjugate A351295.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A051903, A066328, A116861, A130091, A212166, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 18, 28, 47, 71, 108, 166, 252, 382, 587, 869, 1282, 1938, 2832, 4153, 6148, 8962, 12965, 18913, 27301, 39380, 56747, 81226, 115907, 166358, 236000, 334647, 475517, 671806, 947552, 1335679, 1875175, 2630584, 3687589, 5150585, 7183548

Offset: 0

Gus Wiseman, Dec 18 2016

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000009(n). - Seiichi Manyama, Nov 14 2018

			The a(6)=11 twice-partitions are:
((6)), ((5)(1)), ((51)), ((4)(2)), ((42)), ((41)(1)),
((3)(2)(1)), ((31)(2)), ((32)(1)), ((321)), ((21)(2)(1)).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
Gus Wiseman, Sequences enumerating triangles of integer partitions

Cf. A000009, A063834, A270995, A279375, A327553, A327605.

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:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, g(i)*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

nn=20;CoefficientList[Series[Product[(1+PartitionsQ[k]x^k),{k,nn}],{x,0,nn}],x]
(* Second program: *)
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] = If[n > i*(i + 1)/2, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, g[i]*b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

G.f.: Product_{k>0} (1 + A000009(k)*x^k). - Seiichi Manyama, Nov 14 2018

A055887 Number of ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539, 180585336876, 487278670744, 1314838220172

Offset: 0

Christian G. Bower, Jun 09 2000

nonn

Jordan matrices are upper bidiagonal matrices such that (A) the diagonal entries are in sorted order, (B) there are only 1's and 0's on the superdiagonal, (C) for each superdiagonal 1, the two diagonal entries to the left and below it must be equal. Let J(N) be the number of N X N Jordan matrices where the diagonal values are, without loss of generality, taken to be a prefix of some fixed strictly increasing sequence x_1, x_2, x_3, ... If Jordan blocks sorted by eigenvalue with ties broken by block size during the sorting, then J(1, 2, 3, ...) is this sequence. - Warren D. Smith, Jan 28 2002
Number of compositions of n into parts k >= 1 where there are A000041(k) sorts of part k. - Joerg Arndt, Sep 30 2012
Also number of chains of multisets that partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Oct 28 2015
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into constant multisets of a multiset of length n covering an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(3) = 8 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{2},{1,1}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
Factorizations into prime powers, are counted by A000688.
The strongly normal case is A063834.
The strongly normal strict case is A270995.
Twice-partitions of type PPR are counted by A279784, factorizations A295935.
The strict case is A304969.
(End)

			The a(4) = 22 chains of multisets, where notation x-y means "y is a submultiset of x", are: (o-o-o-o) (oo-o-o) (oo-oo) (ooo-o) (oooo) (oe-o-o) (ooe-o) (oooe) (oe-oe) (ooe-e) (oee-o) (ooee) (oei-o) (ooei) (oe-e-e) (oee-e) (oeee) (oei-e) (oeei) (oei-i) (oeii) (oeis).
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose an integer partition of each part of an integer composition of n. The a(0) = 1 through a(3) = 8 choices are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((21))
             ((1)(1))  ((111))
                       ((1)(2))
                       ((2)(1))
                       ((1)(11))
                       ((11)(1))
                       ((1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2320
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89, DOI:10.1007/s11083-004-9460-9.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A055888, A083355, A095975, A246828.
Row sums of A060642.
Cf. A326346.
The unordered version is A001970, row-sums of A061260.
A000041 counts integer partitions, strict A000009.
A011782 counts integer compositions.
A072233 counts partitions by sum and length.
Cf. A001055, A050361, A063834, A279784, A355743.

with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); # F(x) is g.f. for this sequence # Warren D. Smith, Jan 28 2002
A055887rec:= proc(n::integer) local k; option remember; with(combinat): if n = 0 then 1 else add(numbpart(k) *procname(n - k), k=1..n); end if; end proc: seq (A055887rec(n), n=0..10); # Thomas Wieder, Nov 26 2007

a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 23 2010 *)
(1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* Gus Wiseman, Jul 31 2022 *)

Vec(1/(2-1/eta(x+O(x^66)))) \\ Joerg Arndt, Sep 30 2012

Invert transform of partitions numbers A000041.
Let p(k) be the number of integer partitions of k. Furthermore, set a(0)=1. Then a(n) = Sum_{k=1..n} p(k)*a(n-k). - Thomas Wieder, Nov 26 2007
G.f.: 1/( 1 - Sum_{k>=1} p(k)*x^k ) where p(k) = A000041(k) is the number of integer partitions of k. - Joerg Arndt, Sep 30 2012
a(n) ~ c * d^n, where d = 2.698329106474211231263998666188376330713465125913986356769... (see A246828) and c = 0.414113793172792357745578049739573823627306487211379286647... - Vaclav Kotesovec, Mar 29 2014
G.f.: 1/(2 - 1/(x)inf), where (x)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 22 2016

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A364348, A364537, A350845 in not containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  102: {1,2,7}
  105: {2,3,4}
  108: {1,1,2,2,2}

Partitions of this type are counted by A351293, complement A239455.
The conjugate is A351295, union of A048767 (parts A381440, fixed A048768, A217605).
The complement is A381432, union of A381431 (conjugate A351294, parts A381436).
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A181819, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],!MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954

Offset: 0

Vladeta Jovovic, Jun 11 2005

From Gus Wiseman, Sep 13 2022: (Start)
Also the number of multiset partitions of integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,2}}        {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,2}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
Intervals are counted by A001227, ranked by A073485.
The initial version is A007294.
The strict version is A327731.
The version for gapless multisets instead of intervals is A356941.
The case of strict partitions is A356957.
Also the number of multiset partitions of integer partitions of n into distinct constant blocks. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}    {{3}}        {{4}}
         {{1,1}}  {{1,1,1}}    {{2,2}}
                  {{1},{2}}    {{1},{3}}
                  {{1},{1,1}}  {{1,1,1,1}}
                               {{2},{1,1}}
                               {{1},{1,1,1}}
Constant multisets are counted by A000005, ranked by A000961.
The non-strict version is A006171.
The unlabeled version is A089259.
The non-constant block version is A261049.
The version for twice-partitions is A279786, factorizations A296131.
Also the number of multiset partitions of integer partitions of n into constant blocks of odd length. For example, a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,1,1}}      {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
The strict version is A327731 (also).
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A006171, A109386, A219554, A280473, A280486, A288007.
Product_{k>=1} (1 + x^k)^sigma_m(k): this sequence (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
A000041 counts integer partitions, strict A000009.
A000110 counts set partitions.
A072233 counts partitions by sum and length.
Cf. A001055, A001970, A011782, A055887, A061260, A270995, A279784, A294617, A304969.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)
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]]]];
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@chQ/@#&]],{n,0,5}] (* Gus Wiseman, Sep 13 2022 *)

a(n)=polcoeff(prod(k=1,n,prod(j=1,n\k,1+x^(j*k)+x*O(x^n))),n) /* Paul D. Hanna */

N=66;  x='x+O('x^N); gf=1/prod(j=0,N, eta(x^(2*j+1))); gf=prod(j=1,N,(1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

Euler transform of A001227.
Weigh transform of A000005.
G.f. satisfies: log(A(x)) = Sum_{n>=1} A109386(n)/n*x^n, where A109386(n) = Sum_{d|n} d*Sum_{m|d} (m mod 2). - Paul D. Hanna, Jun 26 2005
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). - Paul D. Hanna, Mar 28 2009
G.f.: Product_{n>=1} Q(x^n) where Q(x) is the g.f. of A000009. - Joerg Arndt, Feb 27 2014
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 04 2017
Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - Vaclav Kotesovec, Aug 29 2018

More terms from Paul D. Hanna, Jun 26 2005

A271619 Twice partitioned numbers where the first partition is strict.

Original entry on oeis.org

1, 1, 2, 5, 8, 18, 34, 65, 109, 223, 386, 698, 1241, 2180, 3804, 6788, 11390, 19572, 34063, 56826, 96748, 163511, 272898, 452155, 755928, 1244732, 2054710, 3382147, 5534696, 8992209, 14733292, 23763685, 38430071, 62139578, 99735806, 160183001, 256682598

Offset: 0

Gus Wiseman, Apr 10 2016

nonn

Number of sequences of integer partitions of the parts of some strict partition of n.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000041(n). - Seiichi Manyama, Nov 15 2018

			a(6)=34: {(6);(5)(1),(51);(4)(2),(42);(4)(11),(41)(1),(411);(33);(3)(2)(1),(31)(2),(32)(1),(321);(3)(11)(1),(31)(11),(311)(1),(3111);(22)(2),(222);(21)(2)(1),(22)(11),(211)(2),(221)(1),(2211);(21)(11)(1),(111)(2)(1),(211)(11),(1111)(2),(2111)(1),(21111);(111)(11)(1),(1111)(11),(11111)(1),(111111)}

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)

Cf. A000009, A000041, A063834 (twice partitioned numbers), A270995, A279785, A327552, A327607.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, b(n, i-1) +`if`(i>n, 0,
       b(n-i, i-1)*combinat[numbpart](i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 11 2016

With[{n = 50}, CoefficientList[Series[Product[(1 + PartitionsP[i] x^i), {i, 1, n}], {x, 0, n}], x]]

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

A302478 Products of prime numbers of squarefree index.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 58, 59, 60, 62, 64, 65, 66, 67, 68, 72, 73, 75, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 90, 93, 94

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01:  {}
02:  {{}}
03:  {{1}}
04:  {{},{}}
05:  {{2}}
06:  {{},{1}}
08:  {{},{},{}}
09:  {{1},{1}}
10:  {{},{2}}
11:  {{3}}
12:  {{},{},{1}}
13:  {{1,2}}
15:  {{1},{2}}
16:  {{},{},{},{}}
17:  {{4}}
18:  {{},{1},{1}}
20:  {{},{},{2}}
22:  {{},{3}}
24:  {{},{},{},{1}}
25:  {{2},{2}}
26:  {{},{1,2}}
27:  {{1},{1},{1}}
29:  {{1,3}}
30:  {{},{1},{2}}
31:  {{5}}
32:  {{},{},{},{},{}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A007716, A050320, A056239, A063834, A076610, A270995, A275024, A281113, A296119, A301753, A302242, A302243, A302491.

Select[Range[100],Or[#===1,And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]

ok(n)={!#select(p->!issquarefree(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A000009.
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,1}}    {{1},{2,2,2}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{1},{2},{1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{2},{3},{1,1}}
                                   {{1},{2},{3},{4}}
The non-strict version is A055887.
The strongly normal non-strict version is A063834.
The strongly normal version is A270995.
(End)

			From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:
  ((1))  ((2))     ((3))        ((4))
         ((1)(1))  ((21))       ((31))
                   ((1)(2))     ((1)(3))
                   ((2)(1))     ((2)(2))
                   ((1)(1)(1))  ((3)(1))
                                ((1)(21))
                                ((21)(1))
                                ((1)(1)(2))
                                ((1)(2)(1))
                                ((2)(1)(1))
                                ((1)(1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2816
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for sequences related to partitions
Index entries for sequences related to compositions

Cf. A050342, A055887, A067687, A081362, A279785, A299106.
Row sums of A308680.
The unordered version is A089259, non-strict A001970 (row-sums of A061260).
For partitions instead of compositions we have A270995, non-strict A063834.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by sum and length.
Cf. A279784.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).
G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).
G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).
G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).
a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - Vaclav Kotesovec, May 23 2018

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A381432 Heinz numbers of section-sum partitions. Union of A381431.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A055887 Number of ordered partitions of partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI