A032302 -id:A032302 - 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...

A376944 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)/2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 12, 8, 8, 16, 24, 24, 24, 32, 32, 64, 64, 64, 80, 80, 112, 160, 160, 160, 224, 224, 256, 320, 416, 416, 480, 576, 576, 704, 768, 896, 1152, 1216, 1280, 1536, 1600, 1856, 2112, 2304, 2560, 3200, 3456, 3584, 4224, 4480, 5120, 5760, 6144, 6656, 7808, 9088

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A032302, A053261, A376943, A376945.

nmax = 60; CoefficientList[Series[Sum[2^k * x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^k, {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ sqrt(1 + sqrt(3)) * exp(sqrt((2*log(2)^2 + 2*log(1 - sqrt(3)/2) * log(sqrt(3) - 1) + 4*polylog(2, sqrt(3) - 1) - Pi^2/3)*n)) / (4*3^(1/4)*sqrt(n)).

A376947 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j).

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 6, 6, 10, 10, 14, 14, 26, 26, 38, 46, 58, 66, 86, 94, 130, 146, 182, 214, 274, 306, 382, 438, 530, 602, 750, 838, 1018, 1162, 1390, 1598, 1898, 2154, 2550, 2910, 3402, 3858, 4550, 5134, 5970, 6786, 7846, 8902, 10306, 11618, 13390, 15142, 17346, 19562, 22398

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

In general, if d >= 1, b > 0 and g.f. = Sum_{k>=0} d^k * x^(b*k^2 + c*k) / Product_{j=1..k} (1 - x^j), then a(n) ~ r^c * (b*log(r)^2 + polylog(2, 1-r))^(1/4) * exp(2*sqrt((b*log(r)^2 + polylog(2, 1-r))*n)) / (2*sqrt((2*b*(1-r) + r)*Pi) * n^(3/4)), where r is the smallest positive real root of the equation d*r^(2*b) + r = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003106, A032302, A376943, A376948.

nmax = 80; CoefficientList[Series[Sum[2^k*x^(k*(k+1)) / Product[1-x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ (Pi^2/6 + log(2)^2)^(1/4) * exp(sqrt((Pi^2/3 + 2*log(2)^2)*n)) / (2^(7/4) * sqrt(3*Pi) * n^(3/4)).

A376948 G.f.: Sum_{k>=0} 2^k * x^(k^2) / Product_{j=1..k} (1 - x^j).

Original entry on oeis.org

1, 2, 2, 2, 6, 6, 10, 10, 14, 22, 26, 34, 46, 54, 66, 82, 110, 126, 162, 194, 246, 286, 354, 410, 502, 606, 714, 842, 1014, 1190, 1418, 1658, 1950, 2278, 2666, 3090, 3646, 4198, 4882, 5634, 6558, 7534, 8754, 10002, 11558, 13230, 15218, 17322, 19910, 22702, 25914, 29466, 33606

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003114, A032302, A376945, A376947.

nmax = 80; CoefficientList[Series[Sum[2^k*x^(k^2) / Product[1-x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ (Pi^2/6 + log(2)^2)^(1/4) * exp(sqrt((Pi^2/3 + 2*log(2)^2)*n)) / (2^(3/4) * sqrt(3*Pi) * n^(3/4)).

A266576 Decimal expansion of Pi^2/12 + log(2)^2 + Sum_{j>=1} 1 / (j^2 * 2^(2*j+1)).

Original entry on oeis.org

1, 4, 3, 6, 7, 4, 6, 3, 6, 6, 8, 8, 3, 6, 8, 0, 9, 4, 6, 3, 6, 2, 9, 0, 2, 0, 2, 3, 8, 9, 3, 5, 8, 3, 3, 5, 4, 2, 4, 9, 9, 5, 6, 4, 3, 5, 6, 5, 4, 8, 7, 2, 1, 0, 2, 6, 6, 7, 2, 4, 3, 9, 2, 4, 8, 6, 5, 0, 1, 5, 7, 8, 9, 2, 7, 7, 3, 9, 7, 7, 9, 7, 5, 4, 3, 7, 3, 7, 8, 6, 7, 1, 5, 5, 0, 6, 8, 8, 9, 0, 1, 0, 1, 3, 3

Offset: 1

Vaclav Kotesovec, Jan 04 2016

A constant related to the asymptotics of A032302.

			1.436746366883680946362902023893583354249956435654872102667243924865...

Eric Weisstein's World of Mathematics, Dilogarithm
Eric Weisstein's World of Mathematics, Polylogarithm
Wikipedia, Polylogarithm

Cf. A032302.

evalf(Pi^2/6 + log(2)^2/2 + polylog(2, -1/2), 120);
Digits :=100 ; evalf(dilog(3)) ; # R. J. Mathar, Jan 07 2021

RealDigits[Pi^2/12 + Log[2]^2 + PolyLog[2, 1/4]/2,10,120][[1]]
RealDigits[-PolyLog[2, -2], 10, 120][[1]] (* Vaclav Kotesovec, Jul 29 2019 *)

Pi^2/6 + log(2)^2/2 + polylog(2, -1/2) \\ Michel Marcus, Jan 04 2016

-dilog(-2) \\ Charles R Greathouse IV, Sep 08 2025

Equals Pi^2/12 + log(2)^2 + Sum_{j>=1} 1 / (j^2 * 2^(2*j+1)).
Equals Pi^2/6 + log(2)^2/2 + polylog(2, -1/2).
Equals Pi^2/12 + log(2)^2 + polylog(2, 1/4)/2.
Equals -polylog(2, -2). - Vaclav Kotesovec, Jul 29 2019

A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 6, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 9, 4, 4, 4, 6, 2, 8, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 8, 2, 6, 6

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

Although the average of an empty set is technically indeterminate, we consider it to be distinct from the other subset-averages.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(60) = 9 distinct subset-averages of (3,2,1,1) are 0/0, 1, 4/3, 3/2, 5/3, 7/4, 2, 5/2, 3.

Cf. A032302, A056239, A108917, A275972, A276024, A296150, A299701, A301899, A301957, A316313.

One more than A316314.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Mean/@Subsets[primeMS[n]]]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A316398(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = A056239(d)/bigomega(d)), mapput(m,s,s); k++)); (1+k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316314(n) + 1.

More terms from Antti Karttunen, Sep 23 2018

A366132 Number of unordered pairs of distinct strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 10, 15, 28, 45, 66, 105, 153, 231, 351, 496, 703, 1035, 1431, 2016, 2850, 3916, 5356, 7381, 10011, 13530, 18336, 24531, 32640, 43660, 57630, 75855, 100128, 130816, 170820, 222778, 288420, 372816, 481671, 618828, 793170, 1016025, 1295245

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(3) = 1 through a(8) = 15 pairs of strict partitions:
  {3,21}  {4,31}  {5,32}   {6,42}    {7,43}    {8,53}
                  {5,41}   {6,51}    {7,52}    {8,62}
                  {41,32}  {51,42}   {7,61}    {8,71}
                           {6,321}   {52,43}   {62,53}
                           {42,321}  {61,43}   {71,53}
                           {51,321}  {61,52}   {71,62}
                                     {7,421}   {8,431}
                                     {43,421}  {8,521}
                                     {52,421}  {53,431}
                                     {61,421}  {53,521}
                                               {62,431}
                                               {62,521}
                                               {71,431}
                                               {71,521}
                                               {521,431}

For subsets instead of partitions we have A006516, non-disjoint A003462.
The disjoint case is A108796, non-strict A260669.
For non-strict partitions we have A355389.
The ordered disjoint case is A365662, non-strict A054440.
The ordered version is 2*a(n).
Including equal pairs or twins gives A366317, ordered A304990.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A161680 and A000217 count 2-subsets of {1..n}.
Cf. A000124, A001255, A006827, A032302, A064914, A365661, A365663.

Table[Length[Subsets[Select[IntegerPartitions[n],UnsameQ@@#&],{2}]],{n,0,30}]

a(n) = binomial(A000009(n),2).

A366317 Number of unordered pairs of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 15, 21, 36, 55, 78, 120, 171, 253, 378, 528, 741, 1081, 1485, 2080, 2926, 4005, 5460, 7503, 10153, 13695, 18528, 24753, 32896, 43956, 57970, 76245, 100576, 131328, 171405, 223446, 289180, 373680, 482653, 619941, 794430, 1017451, 1296855

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(1) = 1 through a(7) = 15 unordered pairs of strict partitions:
  {1,1}  {2,2}  {3,3}    {4,4}    {5,5}    {6,6}      {7,7}
                {3,21}   {4,31}   {5,32}   {6,42}     {7,43}
                {21,21}  {31,31}  {5,41}   {6,51}     {7,52}
                                  {32,32}  {42,42}    {7,61}
                                  {32,41}  {42,51}    {43,43}
                                  {41,41}  {51,51}    {43,52}
                                           {6,321}    {43,61}
                                           {42,321}   {52,52}
                                           {51,321}   {52,61}
                                           {321,321}  {61,61}
                                                      {7,421}
                                                      {43,421}
                                                      {52,421}
                                                      {61,421}
                                                      {421,421}

For non-strict partitions we have A086737.
The disjoint case is A108796, non-strict A260669.
The ordered version is A304990, disjoint A032302.
The ordered disjoint case is A365662.
Excluding constant pairs gives A366132.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000712, A007582, A054440, A064914, A260664.

Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2],OrderedQ]],{n,0,30}]

a(n) = A000217(A000009(n)).

Composition of A000009 and A000217.

A370737 a(n) = 5^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/5).

Original entry on oeis.org

1, 10, 50, 14750, -166250, 14011250, -133418750, 18136968750, -620089531250, 29520532031250, -917207280468750, 51260806902343750, -2257145499863281250, 101035630688769531250, -4434459153208496093750, 214279556679692871093750, -9859289197933918457031250, 454976266920750451660156250

Offset: 0

Vaclav Kotesovec, Feb 28 2024

sign

Cf. A032302 (m=1), A370709 (m=2), A370716 (m=3), A370736 (m=4).

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

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

a(n) ~ (-1)^(n+1) * QPochhammer(-1/2)^(1/5) * 50^n / (5 * Gamma(4/5) * n^(6/5)).

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)).

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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A376944 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j).

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A376947 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j).

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A376948 G.f.: Sum_{k>=0} 2^k * x^(k^2) / Product_{j=1..k} (1 - x^j).

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A266576 Decimal expansion of Pi^2/12 + log(2)^2 + Sum_{j>=1} 1 / (j^2 * 2^(2*j+1)).

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PARI

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A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

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Mathematica

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Extensions

A366132 Number of unordered pairs of distinct strict integer partitions of n.

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Mathematica

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A366317 Number of unordered pairs of strict integer partitions of n.

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

A376944 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)/2) Product_{j=1..k} (1 + x^j).

A370737 a(n) = 5^(2n) [x^n] Product_{k>=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).