A304961 -id:A304961 - cp's OEIS frontend

A370765 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/3).

1, 15, 153, 11295, 31968, 5289300, 41957514, 3216919050, -21009764691, 2153132775315, -16978376482767, 1659596014366335, -35929151338082922, 1473739361689662990, -38968782475183427016, 1541715187631618436300, -46858796372722560413526, 1615119529247884664988030

Offset: 0

Vaclav Kotesovec, Mar 01 2024

sign

Cf. A032302, A304961, A370016, A370761, A370764.

nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*(9*x)^k)*(1+2^(k-1)*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, 2*x]*QPochhammer[-1/2, 2*x]/9)^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x]/9)^(1/3), {x, 0, nmax}], x] * 18^Range[0, nmax]

G.f.: Product_{k>=1} ((1 + 2^(k+1)*(9*x)^k) * (1 + 2^(k-1)*(9*x)^k))^(1/3).

a(n) ~ (-1)^(n+1) * c * 36^n / n^(4/3), where c = 0.244280405759762854740979712556383125782589356973734984...

A299786 Expansion of Product_{k>=1} (1 + k^(k-1)*x^k).

Original entry on oeis.org

1, 1, 2, 11, 73, 707, 8547, 127379, 2237804, 45511484, 1049155214, 27060763974, 771662014455, 24109614539775, 818906748562249, 30044648617150066, 1184045057676213763, 49883902402848781573, 2237286132689496359239, 106426356238092171308928, 5352031894869594850387969

Offset: 0

Ilya Gutkovskiy, Jan 21 2019

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * n^(n-1). - Seiichi Manyama, Aug 22 2020

Seiichi Manyama, Table of n, a(n) for n = 0..387

Cf. A265949, A266964, A304961, A321387, A323634.

nmax = 20; CoefficientList[Series[Product[(1 + k^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^(k - k/d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+k^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

a(n) ~ n^(n-1) * (1 + exp(-1)/n + (2*exp(-2) + 3*exp(-1)/2)/n^2). - Vaclav Kotesovec, Jan 22 2019

A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

Original entry on oeis.org

1, 2, 5, 14, 37, 98, 259, 682, 1791, 4697, 12303, 32196, 84199, 220087, 575067, 1502176, 3923117, 10244069, 26746171, 69825070, 182276806, 475804961, 1241965456, 3241732629, 8461261457, 22084402087, 57640875725, 150442742575, 392652788250, 1024810764496

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

By "distinct" we mean equal subsequences are counted only once. For example, the pair (1,1)(1) is counted only once even though (1) is a subsequence of (1,1) in two ways. The version with multiplicity is A025192.

			The a(3) = 14 pairings of a composition with a chosen subsequence:
  (3)()     (3)(3)
  (21)()    (21)(1)   (21)(2)    (21)(21)
  (12)()    (12)(1)   (12)(2)    (12)(12)
  (111)()   (111)(1)  (111)(11)  (111)(111)

Christian Sievers, Table of n, a(n) for n = 0..2000

For partitions we have A000712, composable A339006.
The homogeneous version is A011782, without containment A000302.
With multiplicity we have A025192, for partitions A070933.
The strict case is A032005.
The case of strict subsequences is A236002.
The composable case is A355384, homogeneous without containment A355388.
A075900 counts compositions of each part of a partition.
A304961 counts compositions of each part of a strict partition.
A307068 counts strict compositions of each part of a composition.
A336127 counts compositions of each part of a strict composition.
Cf. A011782, A022811, A032020, A063834, A133494, A181591, A323583, A331330, A336128, A336130, A336139, A355382, A355383.

Table[Sum[Length[Union[Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,6}]

lista(n)=my(f=sum(k=1,n,(x^k+x*O(x^n))/(1-x/(1-x)+x^k)));Vec((1-x)/((1-2*x)*(1-f))) \\ Christian Sievers, May 06 2025

G.f.: (1-x)/((1-2*x)*(1-f)) where f = Sum_{k>=1} x^k/(1-x/(1-x)+x^k) is the generating function for A331330. - Christian Sievers, May 06 2025

a(16) and beyond from Christian Sievers, May 06 2025

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/2).

Original entry on oeis.org

1, 10, 62, 1620, 6966, 157580, 1284012, 19189160, 73908774, 2233414620, 9656822916, 287668788120, -324007115716, 40151699854200, -199460032590312, 7130611518222160, -64971542557275642, 1292318115470489340, -15433712240157937260, 265667290368470451000, -3624776372747687578668

Offset: 0

Vaclav Kotesovec, Mar 01 2024

sign

In general, if d > 1 and g.f. = Product_{k>=1} ((1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k))^(1/2), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d) * d^(2*n) / (2*sqrt((1 + 1/d)*Pi) * n^(3/2)).

Cf. A032302, A304961, A370709, A370761, A370765.

nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(3*k+1)*x^k)*(1+2^(3*k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x])^(1/2)/3, {x, 0, nmax}], x] * 8^Range[0, nmax]

G.f.: Product_{k>=1} ((1 + 2^(3*k+1)*x^k) * (1 + 2^(3*k-1)*x^k))^(1/2).

a(n) ~ (-1)^(n+1) * c * 16^n / n^(3/2), where c = QPochhammer(-1/2) / sqrt(6*Pi) = 0.278865402428524528968820654198674...

A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, -1, -1, -3, -1, -7, -1, -15, 31, -63, 159, -95, 671, -287, 3231, -2975, 15519, -7839, 44191, -34975, 224415, -291999, 863391, -990367, 2927775, -4902047, 12561567, -27225247, 56470687, -102640799, 152153247, -422620319, 877243551, -2278272159, 3357125791

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A032302, A070877, A070933, A071109, A075900, A304961, A327550, A337299, A352762.

nmax = 34; CoefficientList[Series[Product[1/(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 2^(n - k), {k, 0, n}], {n, 0, 34}]

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 2^(n-k), where p(n,k) is the number of partitions of n into k parts.

A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 15, 32, 31, 63, 56, 142, 101, 240, 211, 467, 297, 985, 490, 1524, 1247, 2542, 1255, 6371, 1979, 7486, 7070, 14128, 4565, 32953, 6842, 42229, 37863, 56266, 17887, 192914, 21637, 145820, 197835, 371853, 44583, 772740, 63261, 943966, 1124840

Offset: 0

Gus Wiseman, Dec 14 2022

nonn

			The a(1) = 1 through a(6) = 14 sets:
  {(1)}  {(2)}   {(3)}    {(4)}       {(5)}      {(6)}
         {(11)}  {(21)}   {(22)}      {(32)}     {(33)}
                 {(111)}  {(31)}      {(41)}     {(42)}
                          {(211)}     {(221)}    {(51)}
                          {(1111)}    {(311)}    {(222)}
                          {(2),(11)}  {(2111)}   {(321)}
                                      {(11111)}  {(411)}
                                                 {(2211)}
                                                 {(3111)}
                                                 {(21111)}
                                                 {(111111)}
                                                 {(3),(21)}
                                                 {(3),(111)}
                                                 {(21),(111)}

This is the constant-sum case of A261049, ordered A358906.
The version for all different sums is A271619, ordered A336342.
Allowing repetition gives A305551, ordered A279787.
The version for compositions instead of partitions is A358904.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A098407 counts sets of compositions, ordered A358907.
Cf. A000005, A000041, A038041, A055887, A063834, A074854, A289078, A304961, A305552, A306017.

Table[If[n==0,1,Sum[Binomial[PartitionsP[d],n/d],{d,Divisors[n]}]],{n,0,50}]

a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ Michel Marcus, Dec 14 2022

a(n) = Sum_{d|n} binomial(A000041(d),n/d).

cp's OEIS Frontend

A370765 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/3).

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A299786 Expansion of Product_{k>=1} (1 + k^(k-1)*x^k).

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A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

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A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/2).

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A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

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Mathematica

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A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

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Formula

A370765 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/3).

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/2).