A218482 -id:A218482 - cp's OEIS frontend

A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

1, 1, 3, 6, 13, 24, 49, 91, 179, 341, 664, 1280, 2503, 4872, 9557, 18750, 36927, 72800, 143880, 284660, 564093, 1118911, 2221834, 4415417, 8781591, 17476099, 34799199, 69327512, 138176461, 275503854, 549502119, 1096327380, 2187894634, 4367310138, 8719509111

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(0) = 1 through a(4) = 13 sequences:
  ()  ((1))  ((2))     ((3))        ((4))
             ((11))    ((21))       ((22))
             ((1)(1))  ((111))      ((31))
                       ((1)(2))     ((211))
                       ((2)(1))     ((1111))
                       ((1)(1)(1))  ((1)(3))
                                    ((2)(2))
                                    ((3)(1))
                                    ((11)(11))
                                    ((1)(1)(2))
                                    ((1)(2)(1))
                                    ((2)(1)(1))
                                    ((1)(1)(1)(1))

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

The case of set partitions is A038041.
The version for weakly decreasing lengths is A141199, strictly A358836.
For equal sums instead of lengths we have A279787.
The case of twice-partitions is A306319, distinct A358830.
The unordered version is A319066.
The case of plane partitions is A323429.
The case of constant sums also is A358833.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A000219, A001970, A063834, A218482, A305551, A358835.

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(1 + sum(k=1, n, 1/(1 - polcoef(g, k, y)) - 1))} \\ Andrew Howroyd, Dec 31 2022

G.f.: 1 + Sum_{k>=1} (1/(1 - [y^k]P(x,y)) - 1) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(16) and beyond from Andrew Howroyd, Dec 31 2022

A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 7, 9, 12, 20, 21, 39, 49, 79, 109, 161, 236, 345, 512, 752, 1092, 1628, 2376, 3537, 5171, 7650, 11266, 16634, 24537, 36173, 53377, 78791, 116224, 171598, 253109, 373715, 551434, 814066, 1201466, 1773425, 2617744, 3864050, 5703840, 8419699

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (23)     (33)      (25)       (35)
                    (1111)  (32)     (222)     (52)       (44)
                            (11111)  (111111)  (223)      (53)
                                               (232)      (233)
                                               (322)      (323)
                                               (1111111)  (332)
                                                          (2222)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

The case of partitions is A319169, ranked by A320324.
The weakly decreasing version is A358335, strictly A358901.
For sequences of partitions see A358905.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A358902 = compositions with weakly decreasing A001221, strictly A358903.
A358909 = partitions with weakly decreasing A001222, complement A358910.
Cf. A056239, A063834, A064573, A218482, A279787, A300335, A319066, A319071, A358831, A358908.

b:= proc(n, i) option remember; uses numtheory; `if`(n=0, 1, add(
     (t-> `if`(i<0 or i=t, b(n-j, t), 0))(bigomega(j)), j=1..n))
    end:
a:= n-> b(n, -1):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 12 2024

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,0,10}]

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358907 Number of finite sequences of distinct integer compositions with total sum n.

Original entry on oeis.org

1, 1, 2, 8, 18, 54, 156, 412, 1168, 3200, 8848, 24192, 66632, 181912, 495536, 1354880, 3680352, 9997056, 27093216, 73376512, 198355840, 535319168, 1443042688, 3884515008, 10445579840, 28046885824, 75225974912, 201536064896, 539339293824, 1441781213952

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 18 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((12))     ((13))
                 ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((112))
                 ((2)(1))   ((121))
                 ((1)(11))  ((211))
                 ((11)(1))  ((1111))
                            ((1)(3))
                            ((3)(1))
                            ((1)(12))
                            ((11)(2))
                            ((1)(21))
                            ((12)(1))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))

Alois P. Heinz, Table of n, a(n) for n = 0..1000

For sets instead of sequences we have A098407, partitions A261049.
This is the strict case of A133494.
The case of distinct sums is A336127, constant sums A074854.
The version for sequences of partitions is A358906.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A218482 counts sequences of compositions with weakly decreasing lengths.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all different Omegas.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.

g:= proc(n) option remember; ceil(2^(n-1)) end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
      add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 15 2022

comps[n_]:=Join@@Permutations/@IntegerPartitions[n];
Table[Length[Select[Join@@Table[Tuples[comps/@c],{c,comps[n]}],UnsameQ@@#&]],{n,0,10}]

a(16)-a(29) from Alois P. Heinz, Dec 15 2022

A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 0, 1, -2, 4, -7, 11, -16, 23, -36, 65, -129, 256, -473, 772, -1028, 835, 776, -5755, 17562, -41750, 86678, -165145, 299949, -541837, 1020029, -2068203, 4509512, -10252952, 23465297, -52762788, 115160832, -243018459, 496094524, -982431070, 1894710043, -3574095362

Offset: 0

Ilya Gutkovskiy, Oct 16 2018

After the first term, this is the second term of the n-th differences of A000009, or column n=1 of A378622. - Gus Wiseman, Feb 03 2025

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

Cf. A000593, A320589, A307548.
The version for non-strict partitions is A320590, row n=1 of A175804.
Column n=1 (except first term) of A378622. See also A293467, A377285, A378970, A378971, A380412 (column n=0).
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of strict partitions, differences A129519.
Cf. A000041, A007442, A002865, A008284, A095195, A103446, A116608, A218482, A281425, A320568.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

seq(coeff(series(mul((1+x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 37); # Muniru A Asiru, Oct 16 2018

nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
Prepend[Table[Differences[PartitionsQ/@Range[0,k+1],k][[2]],{k,0,30}],1] (* Gus Wiseman, Jan 29 2025 *)

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, (1 + x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
From Peter Bala, Dec 22 2020: (Start)
O.g.f.: Sum_{n >= 0}  x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
Conjectural o.g.f.: (1/2) * Sum_{n >= 0} x^(n*(n-1)/2)*(1 + x)^n/( Product_{k = 1..n} ( (1 + x)^k - x^k ) ). (End)
a(n+1) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000009(k+1). - Gus Wiseman, Feb 03 2025

A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.

Original entry on oeis.org

1, 1, 2, 5, 11, 23, 49, 103, 214, 434, 874, 1738, 3443, 6765, 13193, 25512, 48957, 93267, 176595, 332550, 622957, 1161230, 2153710, 3974809, 7299707, 13343290, 24280924, 43999100, 79412942, 142792535, 255826836, 456735456, 812627069, 1440971069, 2546729830

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 11 sequences:
  (1)  (2)   (3)      (4)
       (11)  (21)     (22)
             (111)    (31)
             (1)(11)  (211)
             (11)(1)  (1111)
                      (11)(2)
                      (1)(21)
                      (2)(11)
                      (21)(1)
                      (1)(111)
                      (111)(1)

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

The case of set partitions is A007837.
This is the case of A055887 with all distinct lengths.
For distinct sums instead of lengths we have A336342.
The case of twice-partitions is A358830.
The unordered version is A358836.
The version for constant instead of distinct lengths is A358905.
A000041 counts integer partitions, strict A000009.
A063834 counts twice-partitions.
A141199 counts sequences of partitions with weakly decreasing lengths.
Cf. A000219, A001970, A038041, A060642, A218482, A271619, A319066, A358831, A358901, A358906, A358908.

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@Length/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); [subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, 1 + y*polcoef(g, k, y) + O(x*x^n)))]} \\ Andrew Howroyd, Dec 30 2022

Terms a(16) and beyond from Andrew Howroyd, Dec 30 2022

A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 68, 100, 153, 227, 342, 509, 759, 1129, 1678, 2492, 3699, 5477, 8121, 12015, 17795, 26313, 38924, 57541, 85065, 125712, 185758, 274431, 405420, 598815, 884465, 1306165, 1928943, 2848360, 4205979, 6210289, 9169540

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

			The a(0) = 1 through a(6) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (33)
                 (111)  (31)    (32)     (42)
                        (211)   (41)     (51)
                        (1111)  (221)    (222)
                                (311)    (231)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358901.
The version not counting multiplicity is A358902, strict A358903.
The case of partitions is A358909, complement A358910.
The case of equality is A358911, partitions A319169.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A319071, A320324, A358831, A358904, A358908.

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,10}]

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358902 Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 84, 134, 213, 338, 536, 850, 1349, 2136, 3389, 5367, 8509, 13480, 21362, 33843, 53624, 84957, 134600, 213251, 337850, 535251, 847987, 1343440, 2128372, 3371895, 5341977, 8463051, 13407689, 21241181, 33651507, 53312538, 84460690

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (24)
                 (111)  (31)    (32)     (33)
                        (211)   (41)     (42)
                        (1111)  (221)    (51)
                                (311)    (222)
                                (2111)   (231)
                                (11111)  (321)
                                         (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..5004 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358903.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A116608 counts partitions by sum and number of distinct parts.
A334028 counts distinct parts in standard compositions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A046660, A056239, A071625, A129519, A300335, A358831, A358908, A358911.

p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t<=i, b(n-j, t), 0))(p(j)), j=1..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 14 2024

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeNu/@#&]],{n,0,10}]

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The case of compositions is A358335, strictly decreasing A358901.
The complement is counted by A358910.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A320563 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.

Original entry on oeis.org

1, 1, 4, 13, 41, 125, 374, 1103, 3213, 9259, 26430, 74806, 210095, 585890, 1623240, 4470232, 12241799, 33349751, 90410255, 243977941, 655553258, 1754265279, 4676358086, 12420299846, 32873598566, 86721264126, 228051843891, 597905347237, 1563071037798, 4074973824099

Offset: 0

Ilya Gutkovskiy, Oct 15 2018

nonn

First differences of the binomial transform of A000219.

N. J. A. Sloane, Transforms

Cf. A000219, A001157, A103446, A218482, A294500, A320564.

seq(coeff(series(mul((1-x^k/(1-x)^k)^(-k),k=1..n),x,n+1), x, n), n = 0 .. 29); # Muniru A Asiru, Oct 15 2018

nmax = 29; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)).

a(n) ~ Zeta(3)^(7/36) * 2^(n - 11/18) * exp(3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) + (1 - Zeta(3))/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 15 2018

A307310 Expansion of Product_{k>=1} (1 - x^k/(1 - x)^k).

Original entry on oeis.org

1, -1, -2, -3, -4, -4, -1, 9, 34, 89, 200, 409, 779, 1394, 2339, 3624, 4974, 5323, 1682, -13279, -56222, -163136, -408768, -943275, -2059237, -4310179, -8712425, -17072901, -32486302, -60006278, -107341413, -184979170, -303998680, -467127625, -642495990, -696247140

Offset: 0

Ilya Gutkovskiy, Apr 02 2019

sign

First differences of the binomial transform of A010815.

Convolution inverse of A218482.

Cf. A010815, A103446, A129519, A218482, A307311.

a:=series(mul((1-x^k/(1-x)^k),k=1..100),x=0,35): seq(coeff(a,x,n),n=0..34); # Paolo P. Lava, Apr 02 2019

nmax = 35; CoefficientList[Series[Product[(1 - x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

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A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.

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A358907 Number of finite sequences of distinct integer compositions with total sum n.

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A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

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A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.

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A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

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A358902 Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).

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