A304969 -id:A304969 - cp's OEIS frontend

A356930 Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 24, 27, 28, 29, 31, 32, 33, 36, 38, 42, 44, 48, 49, 53, 54, 56, 57, 58, 59, 62, 63, 64, 66, 71, 72, 76, 77, 79, 81, 83, 84, 87, 88, 93, 96, 97, 98, 99, 106, 108, 112, 114, 116, 118, 121, 124, 126, 127

Offset: 1

Gus Wiseman, Sep 11 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The combined size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  24: {{},{},{},{1}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Multisets of odd numbers are counted by A000009, ranked by A066208.
Factorizations of this type are counted by A356931.
The version for odd lengths instead of parts is A356935, ranked by A089259.
Other conditions: A302478, A302492, A356939, A356940, A356944, A356955.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A003963, A011782, A055887, A076610, A302242, A304969.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@(OddQ[Times@@primeMS[#]]&/@primeMS[#])&]

A356933 Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.

Original entry on oeis.org

1, 1, 2, 8, 28, 108, 524, 2608, 14176, 86576, 550672, 3782496, 27843880, 214071392, 1751823600, 15041687664, 134843207240, 1269731540864, 12427331494304, 126619822952928, 1341762163389920, 14712726577081248, 167209881188545344, 1963715680476759040, 23794190474350155856

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

			The a(4) = 28 multiset partitions:
  {1}{111}      {1}{112}      {1}{123}      {1}{234}
  {1}{1}{1}{1}  {1}{122}      {1}{223}      {2}{134}
                {1}{222}      {1}{233}      {3}{124}
                {2}{111}      {2}{113}      {4}{123}
                {2}{112}      {2}{123}      {1}{2}{3}{4}
                {2}{122}      {2}{133}
                {1}{1}{1}{2}  {3}{112}
                {1}{1}{2}{2}  {3}{122}
                {1}{2}{2}{2}  {3}{123}
                              {1}{1}{2}{3}
                              {1}{2}{2}{3}
                              {1}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 0..500
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A055887, A063834, A072233, A270995, A304969, A349050, A349055.
Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other conditions: A034691, A116540, A255906, A356937, A356942.
Other types: A050330, A356932, A356934, A356935.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],OddQ[Times@@Length/@#]&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, if(j%2 == 1, binomial(j+k-1, j))))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Jan 01 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023

A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 32, 36, 39, 48, 52, 54, 64, 72, 78, 81, 96, 104, 108, 113, 117, 128, 144, 156, 162, 169, 192, 208, 216, 226, 234, 243, 256, 288, 312, 324, 338, 339, 351, 384, 416, 432, 452, 468, 486, 507, 512, 576, 624, 648

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An initial interval is a set {1,2,...,n}  for some n >= 0.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  32: {{},{},{},{},{}}
  36: {{},{},{1},{1}}
  39: {{1},{1,2}}
  48: {{},{},{},{},{1}}
  52: {{},{},{1,2}}
  54: {{},{1},{1},{1}}
  64: {{},{},{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

This is the initial version of A356939.
Initial intervals are counted by A010054, ranked by A002110.
Other types: A007294, A322585.
Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A302242, A304969.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
chinQ[y_]:=y==Range[Length[y]];
Select[Range[100],And@@chinQ/@primeMS/@primeMS[#]&]

A356943 Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 4, 11, 37, 101, 328, 909, 2801

Offset: 0

Gus Wiseman, Sep 09 2022

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.

			The a(1) = 1 through a(3) = 11 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Gapless multisets are counted by A034296, ranked by A073491.
Other conditions: A035310, A063834, A330783, A356934, A356938, A356954.
Other types: A356233, A356941, A356942, A356944.
Cf. A055887, A072233, A270995, A304969, A349050, A349055.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[Join@@mps/@strnorm[n],And@@nogapQ/@#&]],{n,0,5}]

A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 26, 27, 28, 32, 36, 37, 38, 39, 42, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 72, 74, 76, 78, 81, 84, 89, 91, 96, 98, 104, 106, 108, 111, 112, 113, 114, 117, 122, 126, 128, 131, 133, 144, 147, 148, 151, 152

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An initial interval is a set {1,2,...,n}  for some n >= 0.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  32: {{},{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Multisets covering an initial interval are ctd by A011782, rkd by A055932.
This is the initial version of A356944.
Other types: A034691, A089259, A356945, A356954.
Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],And@@normQ/@primeMS/@primeMS[#]&]

A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).

Original entry on oeis.org

1, 1, 3, 10, 29, 88, 264, 790, 2366, 7086, 21216, 63523, 190201, 569485, 1705121, 5105383, 15286247, 45769238, 137039743, 410316854, 1228548190, 3678451550, 11013817655, 32976968175, 98737827756, 295635383297, 885175234817, 2650343093602, 7935511791620, 23760073760720, 71141108467679

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

Invert transform of A026007.

a(n) is the number of compositions of n where there are A026007(k) sorts of part k. - Joerg Arndt, Jan 24 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026007, A257674, A299167, A304969.

Cf. A307057, A307058, A307059, A307060, A307063.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+x^j)^j: j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

b:= proc(n) b(n):= add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n)) end:
g:= proc(n) g(n):= `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n) end:
a:= proc(n) a(n):= `if`(n=0, 1, add(g(k)*a(n-k), k=1..n)) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 24 2024

nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/( 2 - product((1+x^j)^j for j in range(1,m+3)) )
def A307062_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307062_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A026007(k)*a(n-k).

A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).

Original entry on oeis.org

1, 1, 3, 10, 28, 85, 252, 745, 2202, 6530, 19326, 57194, 169341, 501242, 1483816, 4392531, 13002772, 38491212, 113943278, 337298400, 998482338, 2955742400, 8749688247, 25901125616, 76673399424, 226971213462, 671887935923, 1988945626648, 5887744768722, 17429103155892, 51594226501776

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

Invert transform of A022629.

a(n) is the number of compositions of n where there are A022629(k) sorts of part k. - Joerg Arndt, Jan 24 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A022629, A299164, A304969, A320652.

Cf. A307057, A307058, A307059, A307060, A307062.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+j*x^j): j in [1..m+2]])) ));

nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/( 2 - product(1+j*x^j for j in range(1,m+3)) )
def A307063_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307063_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A022629(k)*a(n-k).

A326346 Total number of partitions in the partitions of compositions of n.

Original entry on oeis.org

0, 1, 4, 14, 47, 151, 474, 1457, 4414, 13210, 39155, 115120, 336183, 976070, 2819785, 8110657, 23239662, 66362960, 188930728, 536407146, 1519205230, 4293061640, 12106883585, 34079016842, 95762829405, 268670620736, 752676269695, 2105751165046, 5883798478398

Offset: 0

Alois P. Heinz, Sep 11 2019

nonn

			a(3) = 14 = 1+1+1+2+2+2+2+3 counts the partitions in 3, 21, 111, 2|1, 11|1, 1|2, 1|11, 1|1|1.

Alois P. Heinz, Table of n, a(n) for n = 0..2313

Cf. A000041, A030267, A055887, A060642, A304969, A325915, A327548.

b:= proc(n) option remember; `if`(n=0, [1, 0], (p-> p+
      [0, p[1]])(add(combinat[numbpart](j)*b(n-j), j=1..n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..32);

b[n_] := b[n] = If[n==0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[ PartitionsP[j] b[n-j], {j, 1, n}]]];
a[n_] := b[n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A060642(n,k).

a(n) ~ c * d^n * n, where d = A246828 = 2.69832910647421123126399866618837633... and c = 0.171490233695958246364725709205670983251448838158816... - Vaclav Kotesovec, Sep 14 2019

A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 22, 41, 72, 142, 260, 454, 769, 1416, 2472, 4465, 7708, 13314, 23630, 40406, 68196, 119646, 203237, 343242, 586508, 993764, 1677187, 2824072, 4753066, 7934268, 13355658, 22229194, 36945828, 61555136, 102019156, 168474033, 279181966

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (1,2)    (1,3)      (1,4)
            (2,1)    (3,1)      (2,3)
            (2),(1)  (3),(1)    (3,2)
                     (1,2),(1)  (4,1)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1,2),(2)
                                (1,3),(1)
                                (2,1),(2)
                                (3,1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..7725

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020

strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n == 0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0,
     If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

G.f.: Product_{k >= 1} (1 + A032020(k)*x^k).

A356954 Number of multisets of multisets, each covering an initial interval, whose multiset union is of size n and has weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 3, 6, 15, 30, 71, 145, 325, 680

Offset: 0

Gus Wiseman, Sep 09 2022

			The a(1) = 1 through a(4) = 15 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
         {{1},{1}}  {{1,2,3}}      {{1,1,2,2}}
                    {{1},{1,1}}    {{1,1,2,3}}
                    {{1},{1,2}}    {{1,2,3,4}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1,1},{1,1}}
                                   {{1},{1,1,2}}
                                   {{1,1},{1,2}}
                                   {{1},{1,2,2}}
                                   {{1},{1,2,3}}
                                   {{1,2},{1,2}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{1,2}}
                                   {{1},{1},{1},{1}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

For unrestricted multiplicities we have A034691.
A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Other conditions: A035310, A063834, A330783, A356934, A356938, A356943.
Other types: A055932, A089259, A356945, A356955.
Cf. A055887, A072233, A270995, A304969, A349050, A349055, A356942.

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]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],And@@normQ/@#&]],{n,0,5}]

cp's OEIS Frontend

A356930 Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.

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A356933 Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.

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PARI

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A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).

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Mathematica

A356943 Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.

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Mathematica

A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

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Mathematica

A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).

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Maple

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SageMath

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A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).

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SageMath

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A326346 Total number of partitions in the partitions of compositions of n.

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