A070933 -id:A070933 - cp's OEIS frontend

A266945 Expansion of Product_{k>=1} 1 / (1 - 2*x^k)^3.

1, 6, 30, 122, 450, 1518, 4830, 14586, 42330, 118622, 322974, 857298, 2226586, 5672046, 14205654, 35040722, 85269114, 204971478, 487307542, 1146995154, 2675265522, 6188176838, 14205568950, 32383725450, 73352114450, 165171276822, 369904716750, 824244212554

Offset: 0

Vaclav Kotesovec, Jan 06 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A070933, A266943.

nmax = 40; CoefficientList[Series[Product[1/(1-2*x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * n^2 * 2^n, where c = 1/(2*A048651^3) = 1/(2*QPochhammer(1/2)^3) = 20.760229307499152409838537... .

A278464 Total number of parts of the second sort in all partitions of n into two sorts of parts.

Original entry on oeis.org

0, 1, 5, 17, 53, 145, 385, 957, 2333, 5493, 12741, 28941, 65049, 144225, 317229, 691457, 1497901, 3224145, 6906969, 14726701, 31282421, 66211253, 139720445, 294007373, 617154865, 1292516577, 2701451621, 5635565761, 11736442005, 24403092657, 50666528209

Offset: 0

Alois P. Heinz, Nov 22 2016

nonn

a(n) is odd for n > 0.

Alois P. Heinz, Table of n, a(n) for n = 0..3309
William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], Feb 2017

Cf. A006128, A070933, A256193.

b:= proc(n, i) option remember; `if`(n=0, [1/2, 0], `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, (p-> p+[0, p[1]])(2*b(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..35);

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*Sum[x^t*Binomial[j, t], {t, 0, j}], {j, 0, n/i}]]]];
a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]] . Range[0, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} k * A256193(n,k).

A349921 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 2 * k^(-s)).

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 14, 6, 6, 2, 18, 2, 6, 6, 34, 2, 18, 2, 18, 6, 6, 2, 46, 6, 6, 14, 18, 2, 22, 2, 74, 6, 6, 6, 58, 2, 6, 6, 46, 2, 22, 2, 18, 18, 6, 2, 114, 6, 18, 6, 18, 2, 46, 6, 46, 6, 6, 2, 70, 2, 6, 18, 166, 6, 22, 2, 18, 6, 22, 2, 150, 2, 6, 18

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

nonn

Cf. A001055, A061142, A070933, A301830, A349922.

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

A370733 a(n) = 5^(2n) [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/5).

Original entry on oeis.org

1, 10, 550, 19750, 921250, 32011250, 1563143750, 58080093750, 2719958906250, 113913469531250, 5214823539843750, 228024893230468750, 10704801509316406250, 482674223446582031250, 22664252188144042968750, 1053427002068999511718750, 49776941230938518066406250

Offset: 0

Vaclav Kotesovec, Feb 28 2024

nonn

Cf. A070933 (m=1), A370713 (m=2), A370715 (m=3), A370732 (m=4).

nmax = 20; CoefficientList[Series[Product[1/(1-2*x^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-2*(25*x)^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - 2*(25*x)^k)^(1/5).

a(n) ~ 50^n / (Gamma(1/5) * QPochhammer(1/2)^(1/5) * n^(4/5)).

A370749 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^(1/4).

Original entry on oeis.org

1, 2, 6, 28, 70, 300, 892, 3544, 9990, 43340, 127988, 546120, 1651356, 7227896, 22414008, 99344944, 312879302, 1396285452, 4486205892, 20057934312, 65293087284, 292353604136, 963327294536, 4308913730256, 14340603113372, 64059675491512, 215075154021384, 958968160741328

Offset: 0

Vaclav Kotesovec, Feb 29 2024

nonn

Cf. A261584, A303346, A370750.
Cf. A032302, A070933.
Cf. A370736, A370732.

nmax = 30; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 2*(2*x)^k)/(1 - 2*(2*x)^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + 2*(2*x)^k)/(1 - 2*(2*x)^k))^(1/4).

a(n) ~ QPochhammer(-1, 1/2)^(1/4) * 4^n / (Gamma(1/4) * QPochhammer(1/2)^(1/4) * n^(3/4)).

A370750 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^(1/3).

Original entry on oeis.org

1, 12, 180, 3852, 50436, 947052, 14087844, 245858652, 3531115620, 64019229660, 950199749748, 16959724619004, 256888616329044, 4642974930688812, 71716402072904724, 1308491345357401068, 20501966472318764388, 376230182366985289164, 5987314157007778195716, 110286515004790197907836

Offset: 0

Vaclav Kotesovec, Feb 29 2024

nonn

Cf. A261584, A303346, A370749.
Cf. A032302, A070933.
Cf. A370716, A370715.

nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[(1 + 2*(9*x)^k)/(1 - 2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

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

a(n) ~ QPochhammer(-1, 1/2)^(1/3) * 18^n / (Gamma(1/3) * QPochhammer(1/2)^(1/3) * n^(2/3)).

A255180 Number of partitions of n in which two summands (of each size) are designated.

Original entry on oeis.org

1, 0, 1, 3, 7, 10, 20, 24, 45, 61, 103, 140, 246, 325, 517, 728, 1086, 1472, 2184, 2918, 4197, 5638, 7875, 10497, 14625, 19272, 26354, 34804, 46992, 61490, 82471, 107163, 142128, 184141, 241701, 311282, 406164, 519755, 672726, 858110, 1102872

Offset: 0

Geoffrey Critzer, Mar 19 2015

nonn

			a(4)=7. In order to designate two summands of each size, the multiplicity of each summand must be at least two. For n=4 we consider only the partitions 2+2 and 1+1+1+1.  In the first case there is 1 way and in the second case there are 6 ways.  1 + 6 = 7.

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

Cf. A077285, A070933 (where any number of summands of any size are designated).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*binomial(j, 2), j=2..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2015

nn = 40; CoefficientList[Series[Product[1 + x^(2 n)/(1 - x^n)^3, {n, 1, nn}], {x, 0, nn}], x]

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

A291583 Expansion of Product_{k>=1} 1/(1 - 2*x^(k^2)).

Original entry on oeis.org

1, 2, 4, 8, 18, 36, 72, 144, 292, 586, 1172, 2344, 4696, 9396, 18792, 37584, 75186, 150380, 300764, 601528, 1203092, 2406200, 4812408, 9624816, 19249704, 38499446, 76998908, 153997824, 307995792, 615991660, 1231983352, 2463966720, 4927933732, 9855867616

Offset: 0

Vaclav Kotesovec, Aug 31 2017

nonn

Cf. A001156, A070933.

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

a(n) = c * 2^n, where c = Product_{k>=2} 1/(1 - 2^(1-k^2)) = 1.1473740190706250791012378892728990358979800885299281286906257674366893358222...

A300582 Expansion of Product_{k>=1} 1 / (1 - 32^kx^k).

Original entry on oeis.org

1, 6, 48, 312, 2064, 12768, 79680, 484224, 2947584, 17783808, 107268096, 644960256, 3877367808, 23282294784, 139790696448, 838984925184, 5035133042688, 30213857476608, 181298502303744, 1087829443608576, 6527166069080064, 39163476131708928, 234983177934864384

Offset: 0

Vaclav Kotesovec, Mar 09 2018

nonn

Cf. A070933, A300583.

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

a(n) ~ c * 6^n, where c = 1/QPochhammer(1/3) = 1.7853123419985341903674862960137...

cp's OEIS Frontend

A266945 Expansion of Product_{k>=1} 1 / (1 - 2*x^k)^3.

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A278464 Total number of parts of the second sort in all partitions of n into two sorts of parts.

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Maple

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A349921 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 2 * k^(-s)).

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

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A370733 a(n) = 5^(2*n) * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/5).

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A370749 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^(1/4).

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A370750 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^(1/3).

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Crossrefs

Programs

Mathematica

Formula

A255180 Number of partitions of n in which two summands (of each size) are designated.

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Examples

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Maple

Mathematica

Formula

A291583 Expansion of Product_{k>=1} 1/(1 - 2*x^(k^2)).

Views

Author

Keywords

Crossrefs

Programs

A370733 a(n) = 5^(2n) [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/5).

A370749 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^(1/4).

A370750 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^(1/3).

A300582 Expansion of Product_{k>=1} 1 / (1 - 32^kx^k).