A063834 -id:A063834 - cp's OEIS frontend

A358824 Number of twice-partitions of n of odd length.

0, 1, 2, 4, 7, 15, 32, 61, 121, 260, 498, 967, 1890, 3603, 6839, 12972, 23883, 44636, 82705, 150904, 275635, 501737, 905498, 1628293, 2922580, 5224991, 9296414, 16482995, 29125140, 51287098, 90171414, 157704275, 275419984, 479683837, 833154673, 1442550486, 2493570655

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             (1)(1)(1)  (211)       (221)
                        (1111)      (311)
                        (2)(1)(1)   (2111)
                        (11)(1)(1)  (11111)
                                    (2)(2)(1)
                                    (3)(1)(1)
                                    (11)(2)(1)
                                    (2)(11)(1)
                                    (21)(1)(1)
                                    (11)(11)(1)
                                    (111)(1)(1)
                                    (1)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for set partitions is A024429.
For odd lengths (instead of length) we have A358334.
The case of odd parts also is A358823.
The case of odd sums also is A358826.
The case of odd lengths also is A358834.
For multiset partitions of integer partitions: A358837, ranked by A026424.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A270995, A271619, A279374, A279785, A306319, A336342, A356932.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&]],{n,0,10}]

R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=vector(n,k,numbpart(k))); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

G.f.: ((1/Product_{k>=1} (1-A000041(k)*x^k)) - (1/Product_{k>=1} (1+A000041(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 30 2022

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 2025

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.

The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[IntegerPartitions/@prix[n]],UnsameQ@@#&]],{n,100}]

A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 12, 13, 35, 37, 98, 107, 304, 336, 927, 1037, 3010, 3367, 9585, 10924, 32126, 36438, 105589, 121045, 359691, 412789, 1211214, 1398168, 4188930, 4831708, 14315544, 16636297, 50079792, 58084208, 173370663, 202101971, 611487744, 712709423

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd p-tree of weight n > 0 is either a single node (if n = 1) or a finite sequence of at least 3 odd p-trees whose weights are weakly decreasing odd numbers summing to n.

			The a(7) = 5 odd p-trees: ((ooo)(ooo)o), (((ooo)oo)oo), ((ooooo)oo), ((ooo)oooo), (ooooooo).

Cf. A000009, A027193, A063834, A078408, A196545, A279374, A279785, A289501, A298118, A299202, A299203, A300300, A300301, A300355, A300439, A300440.

b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&And@@OddQ/@#&]}];
Table[b[n],{n,40}]

O.g.f: x + Product_{n odd} 1/(1 - a(n)*x^n) - Sum_{n odd} a(n)*x^n. - Gus Wiseman, Aug 27 2018

Name corrected by Gus Wiseman, Aug 27 2018

A301364 Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 4, 5, 1, 2, 6, 11, 12, 1, 3, 10, 26, 38, 34, 1, 3, 13, 39, 87, 117, 92, 1, 4, 19, 69, 181, 339, 406, 277, 1, 4, 23, 95, 303, 707, 1198, 1311, 806, 1, 5, 30, 143, 514, 1430, 2970, 4525, 4522, 2500, 1, 5, 35, 184, 762, 2446, 6124, 11627

Offset: 1

Gus Wiseman, Mar 19 2018

An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			Triangle begins:
  1
  1   1
  1   1   2
  1   2   4   5
  1   2   6  11  12
  1   3  10  26  38  34
  1   3  13  39  87 117  92
  1   4  19  69 181 339 406 277
  ...
The T(5,4) = 11 enriched p-trees: (((21)1)1), ((2(11))1), (((11)2)1), ((211)1), ((21)(11)), (((11)1)2), ((111)2), ((21)11), (2(11)1), ((11)21), (2111).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Last entries of each row give A196545. Row sums are A289501.

Cf. A008284, A055277, A063834, A220418, A273866, A273873, A281145, A290261, A299201, A299202, A299203, A300354, A300442, A300443, A301365-A301368.

eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[Select[eptrees[n],Count[#,_Integer,{-1}]===k&]],{n,8},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

A306319 Number of length-rectangular twice-partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 60, 82, 131, 177, 286, 376, 582, 793, 1202, 1610, 2450, 3274, 4906, 6665, 9770, 13274, 19690, 26506, 38596, 53006, 76432, 104189, 150844, 205282, 294304, 404146, 573140, 786169, 1119457, 1527554, 2155953, 2965567, 4163955, 5701816

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is length-rectangular if all parts have the same number of parts.

			The a(5) = 14 length-rectangular twice-partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Dominates A319066 (rectangular partitions of partitions), which dominates A323429 (rectangular plane partitions).

Cf. A000219, A001970, A063834 (twice-partitions), A089299, A271619, A279787 (sum-rectangular twice-partitions), A305551, A306017, A306318 (square case), A323531.

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],SameQ@@Length/@#&],{ptn,IntegerPartitions[n]}]],{n,20}]

A307068 Expansion of 1/(1 - Sum_{k>=1} k!x^(k(k+1)/2) / Product_{j=1..k} (1 - x^j)).

Original entry on oeis.org

1, 1, 2, 6, 14, 34, 88, 216, 532, 1322, 3290, 8142, 20192, 50080, 124144, 307878, 763474, 1893038, 4694060, 11639580, 28861736, 71567206, 177460750, 440037738, 1091134276, 2705618900, 6708953156, 16635775698, 41250705518, 102286806130, 253634237896, 628921097352, 1559496588628

Offset: 0

Ilya Gutkovskiy, Mar 22 2019

nonn

Invert transform of A032020.
Number of ways to choose a strict composition of each part of a composition of n. - Gus Wiseman, Jul 18 2020
The Invert transform T(a) of a sequence a is given by T(a)n = Sum_c Product_i a(c_i), where the sum is over all compositions c of n. - _Gus Wiseman, Aug 01 2020

			From _Gus Wiseman_, Jul 18 2020: (Start)
The a(1) = 1 through a(4) = 14 ways to choose a strict composition of each part of a composition:
    (1)  (2)      (3)          (4)
         (1),(1)  (1,2)        (1,3)
                  (2,1)        (3,1)
                  (1),(2)      (1),(3)
                  (2),(1)      (2),(2)
                  (1),(1),(1)  (3),(1)
                               (1),(1,2)
                               (1),(2,1)
                               (1,2),(1)
                               (2,1),(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..2536

The version for partitions is A270995.
Starting with a strict composition gives A336139.
Strict compositions are counted by A032020.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Compositions of each part of a composition are A133494.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict partitions of each part of a composition are A304969.
Compositions of each part of a strict composition are A336127.
Set partitions of strict compositions are A336140.
Strict compositions of each part of a partition are A336141.
Cf. A001970, A307067, A317536, A318683, A318684, A319794, A323583, A336128, A336130, A336132.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(1 - (&+[Factorial(k)*x^Binomial(k+1,2)/(&*[ 1-x^j: j in [1..k]]): k in [1..m+2]]) ) )); // G. C. Greubel, Jan 25 2024

T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
    end:
g:= proc(n) option remember; add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)) end:
a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*g(i), i=1..n))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 16 2022

nmax = 32; CoefficientList[Series[1/(1 - Sum[k!*x^(k*(k+1)/2)/Product[ (1-x^j), {j,k}], {k,nmax}]), {x, 0, nmax}], x]

m=80;
def p(x, j): return product(1-x^k for k in range(1,j+1))
def f(x): return 1/(1 - sum(factorial(j)*x^binomial(j+1,2)/p(x,j) for j in range(1, m+3)) )
def A307068_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307068_list(m) # G. C. Greubel, Jan 25 2024

a(0) = 1; a(n) = Sum_{k=1..n} A032020(k)*a(n-k).

A323718 Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 6, 4, 1, 1, 1, 7, 15, 10, 5, 1, 1, 1, 11, 28, 34, 15, 6, 1, 1, 1, 15, 66, 80, 65, 21, 7, 1, 1, 1, 22, 122, 254, 185, 111, 28, 8, 1, 1, 1, 30, 266, 604, 739, 371, 175, 36, 9, 1, 1, 1, 42, 503, 1785, 2163, 1785, 672, 260, 45, 10, 1, 1

Offset: 0

Gus Wiseman, Jan 25 2019

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n, and the only 0-times partition of n is the number n itself.

			Array begins:
       k=0:   k=1:   k=2:   k=3:   k=4:   k=5:
  n=0:  1      1      1      1      1      1
  n=1:  1      1      1      1      1      1
  n=2:  1      2      3      4      5      6
  n=3:  1      3      6     10     15     21
  n=4:  1      5     15     34     65    111
  n=5:  1      7     28     80    185    371
  n=6:  1     11     66    254    739   1785
  n=7:  1     15    122    604   2163   6223
  n=8:  1     22    266   1785   8120  28413
  n=9:  1     30    503   4370  24446 101534
The A(4,2) = 15 twice-partitions:
  (4)  (31)    (22)    (211)      (1111)
       (3)(1)  (2)(2)  (11)(2)    (11)(11)
                       (2)(11)    (111)(1)
                       (21)(1)    (11)(1)(1)
                       (2)(1)(1)  (1)(1)(1)(1)

Alois P. Heinz, Rows n = 0..140, flattened

Columns: A000012 (k=0), A000041 (k=1), A063834 (k=2), A301595 (k=3).
Rows: A000027 (n=2), A000217 (n=3), A006003 (n=4).
Main diagonal gives A306187.
Cf. A001970, A055884, A096751, A144150, A196545, A281113, A289501, A290353, A300383, A323719, A327618, A327639.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Jan 25 2019

ptnlev[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Tuples[ptnlev[#,k-1]&/@ptn],{ptn,IntegerPartitions[n]}]];
Table[Length[ptnlev[sum-k,k]],{sum,0,12},{k,0,sum}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1, 1,
     b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
A[n_, k_] := b[n, n, k];
Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

Column k is the formal power product transform of column k-1, where the formal power product transform of a sequence q with offset 1 is the sequence whose ordinary generating function is Product_{n >= 1} 1/(1 - q(n) * x^n).

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327639(n,i). - Alois P. Heinz, Sep 20 2019

A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 4, 12, 30, 66, 164, 419, 1049, 2625, 6372, 15451, 37335, 89855, 216523, 518714, 1235897, 2930050, 6911149, 16217817, 37914515, 88304358, 204971388, 474172899, 1093547574, 2513959446, 5761735383, 13165908506, 29998936859, 68164839887, 154478212575

Offset: 0

Gus Wiseman, Jul 01 2022

nonn

If a composition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of compositions.

			The initial terms count the following containments:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(1)  (21)(21)  (31)(31)
                         (12)(12)  (13)(13)
                         (111)(1)  (22)(2)
                                   (211)(11)
                                   (211)(21)
                                   (121)(11)
                                   (121)(12)
                                   (121)(21)
                                   (112)(11)
                                   (112)(12)
                                   (1111)(1)

Christian Sievers, Table of n, a(n) for n = 0..59

The homog. case is A032020, w/o containment A355388 (partitions A355385).
For partitions we have A355383, homog. A000009, w/ multiplicity A339006.
A000244 counts splittings of compositions, for partitions A323583.
Cf. A001970, A022811, A063834, A070933, A072706, A133494, A336139.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,5}]

a(21) and beyond from Christian Sievers, May 08 2025

A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 5, 8, 7, 6, 6, 16, 11, 8, 13, 12, 6, 10, 19, 16, 9, 14, 8, 12, 29, 12, 37, 32, 10, 22, 9, 16, 47, 26, 14, 24, 61, 12, 79, 20, 12, 38, 103, 32, 9, 18, 22, 28, 131, 16, 15, 24, 26, 58, 163, 24, 199, 74, 12, 64, 21, 20, 251, 44, 38

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

The non-strict version is A357977.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357979, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsQ],100]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f9(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A358334 Number of twice-partitions of n into odd-length partitions.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765

Offset: 0

Gus Wiseman, Dec 01 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(0) = 1 through a(5) = 13 twice-partitions:
  ()  ((1))  ((2))     ((3))        ((4))           ((5))
             ((1)(1))  ((111))      ((211))         ((221))
                       ((2)(1))     ((2)(2))        ((311))
                       ((1)(1)(1))  ((3)(1))        ((3)(2))
                                    ((111)(1))      ((4)(1))
                                    ((2)(1)(1))     ((11111))
                                    ((1)(1)(1)(1))  ((111)(2))
                                                    ((211)(1))
                                                    ((2)(2)(1))
                                                    ((3)(1)(1))
                                                    ((111)(1)(1))
                                                    ((2)(1)(1)(1))
                                                    ((1)(1)(1)(1)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For multiset partitions of integer partitions: A356932, ranked by A356935.
For odd length instead of lengths we have A358824.
For odd sums instead of lengths we have A358825.
For odd sums also we have A358827.
For odd length also we have A358834.
A000041 counts integer partitions.
A027193 counts odd-length partitions, ranked by A026424.
A055922 counts partitions with odd multiplicities, also odd parts A117958.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1), -(n+1))} \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - A027193(k)*x^k). - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

cp's OEIS Frontend

A358824 Number of twice-partitions of n of odd length.

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A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

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A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

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A301364 Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.

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A306319 Number of length-rectangular twice-partitions of n.

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A307068 Expansion of 1/(1 - Sum_{k>=1} k!*x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)).

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A323718 Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.

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A307068 Expansion of 1/(1 - Sum_{k>=1} k!x^(k(k+1)/2) / Product_{j=1..k} (1 - x^j)).