A075900 -id:A075900 - cp's OEIS frontend

A338697 a(n) = [x^n] Product_{k>=1} 1 / (1 - n^(k-1)*x^k).

1, 1, 3, 13, 101, 931, 12391, 178809, 3331721, 66288142, 1589753211, 40104031166, 1183380156013, 36187564837217, 1262524447510383, 45533370885563716, 1834219414937219601, 76016894083755947753, 3479900167920331954531, 162982921698852088968886, 8341707623665223127224821

Offset: 0

Ilya Gutkovskiy, Apr 24 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

Cf. A008284, A075900, A124577, A300579, A338673, A338674, A338675, A338676, A338677, A338678, A338679, A344095.

Table[SeriesCoefficient[Product[1/(1 - n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Length[IntegerPartitions[n, {k}]] n^(n - k), {k, 0, n}], {n, 1, 20}]]
Join[{1}, Table[SeriesCoefficient[x + (n-1)/(n*QPochhammer[1/n, n*x]), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)

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

a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - Vaclav Kotesovec, May 09 2021

A370337 Expansion of Product_{n>=1} (1 - 2^(n-1)x^n) (1 + 2^(n-1)*x^n)^2.

Original entry on oeis.org

1, 1, 1, 5, 6, 22, 40, 108, 192, 536, 1072, 2528, 5344, 12288, 26624, 61312, 129024, 286720, 646656, 1389568, 3028992, 6717440, 14708736, 31604736, 69763072, 150110208, 329809920, 714473472, 1546649600, 3324772352, 7332954112, 15626403840, 33840693248, 73194799104, 158456610816

Offset: 0

Paul D. Hanna, Feb 26 2024

nonn

Compare to Product_{n>=1} (1 - 2^n*x^n) * (1 + 2^n*x^n)^2 = Sum_{n>=0} 2^(n*(n+1)/2) * x^(n*(n+1)/2).

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 6*x^4 + 22*x^5 + 40*x^6 + 108*x^7 + 192*x^8 + 536*x^9 + 1072*x^10 + 2528*x^11 + 5344*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 2*x^2)*(1 + 2*x^2)^2 * (1 - 4*x^3)*(1 + 4*x^3)^2 * (1 - 8*x^4)*(1 + 8*x^4)^2 * ... * (1 - 2^(n-1)*x^n)*(1 + 2^(n-1)*x^n)^2 * ...
Compare A(x) to the series that results from a similar infinite product:
(1 - 2*x)*(1 + 2*x)^2 * (1 - 4*x^2)*(1 + 4*x^2)^2 * (1 - 8*x^3)*(1 + 8*x^3)^2 * (1 - 16*x^4)*(1 + 16*x^4)^2 * ... = 1 + 2*x + 8*x^3 + 64*x^6 + 1024*x^10 + 32768*x^15 + 2097152*x^21 + 268435456*x^28 + 68719476736*x^36 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1035

Cf. A006125, A075900, A304961, A337299, A355234, A370338.

CoefficientList[Series[8*QPochhammer[1/2, 2*x] * QPochhammer[-1/2, 2*x]^2/9, {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 26 2024 *)

{a(n) = polcoeff( prod(k=1,n, (1 - 2^(k-1)*x^k) * (1 + 2^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,40, print1(a(n),", "))

a(n) ~ c^(1/4) * 2^n * exp(sqrt(c*n)) / (3*sqrt(Pi)*n^(3/4)), where c = 2*log(2)^2 - Pi^2/3 - 8*polylog(2,-1/2) = Pi^2 + 6*log(2)^2 + 8*polylog(2,-2) = 1.258351549529119595933889966687474131697... - Vaclav Kotesovec, Feb 26 2024

A358904 Number of finite sets of compositions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 38, 64, 156, 260, 632, 1024, 2601, 4096, 10208, 16944, 40966, 65536, 168672, 262144, 656980, 1090240, 2620928, 4194304, 10862100, 16781584, 41940992, 69872384, 168403448, 268435456, 693528552, 1073741824, 2695006177, 4473400320, 10737385472

Offset: 0

Gus Wiseman, Dec 13 2022

nonn

			The a(1) = 1 through a(4) = 9 sets:
  {(1)}  {(2)}   {(3)}    {(4)}
         {(11)}  {(12)}   {(13)}
                 {(21)}   {(22)}
                 {(111)}  {(31)}
                          {(112)}
                          {(121)}
                          {(211)}
                          {(1111)}
                          {(2),(11)}

This is the constant-sum case of A098407, ordered A358907.
The version for distinct sums is A304961, ordered A336127.
Allowing repetition gives A305552, ordered A074854.
The case of sets of partitions is A359041.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A261049 counts sets of partitions, ordered A358906.
Cf. A000009, A063834, A075900, A218482, A296122.

Table[If[n==0,1,Sum[Binomial[2^(d-1),n/d],{d,Divisors[n]}]],{n,0,30}]

a(n) = if (n, sumdiv(n, d, binomial(2^(d-1), n/d)), 1); \\ Michel Marcus, Dec 14 2022

a(n>0) = Sum_{d|n} binomial(2^(d-1),n/d).

A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

Original entry on oeis.org

1, 4, 24, 112, 544, 2368, 10624, 44800, 190976, 791552, 3282944, 13414400, 54829056, 222117888, 899383296, 3625123840, 14601027584, 58659700736, 235555782656, 944552017920, 3786334535680, 15166305468416, 60736264994816, 243129089261568, 973133053952000

Offset: 0

Vaclav Kotesovec, Mar 09 2018

nonn

Cf. A075900, A300579.

Cf. A023882, A077365, A179381, A300520.

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

a(n) ~ c * 4^n, where c = A065446 = 1/QPochhammer(1/2) = 3.46274661945506361...

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

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.

cp's OEIS Frontend

A338697 a(n) = [x^n] Product_{k>=1} 1 / (1 - n^(k-1)*x^k).

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A370337 Expansion of Product_{n>=1} (1 - 2^(n-1)*x^n) * (1 + 2^(n-1)*x^n)^2.

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A358904 Number of finite sets of compositions with all equal sums and total sum n.

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A300581 Expansion of Product_{k>=1} 1/(1 - 2^(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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A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

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A370337 Expansion of Product_{n>=1} (1 - 2^(n-1)x^n) (1 + 2^(n-1)*x^n)^2.