A338673 -id:A338673 - cp's OEIS frontend

A300579 Expansion of Product_{k>=1} 1/(1 - 3^(k-1)*x^k).

1, 1, 4, 13, 49, 157, 589, 1885, 6826, 22378, 78754, 256630, 904711, 2934247, 10133851, 33287620, 113522089, 370582069, 1262300701, 4110883510, 13869616495, 45364050184, 151708228636, 494743296757, 1654133919475, 5379427446952, 17858926956532, 58219580395822

Offset: 0

Vaclav Kotesovec, Mar 09 2018

nonn

In general, if g.f. = Product_{k>=1} 1/(1 - d^(k-1)*x^k), where d > 1, then a(n) ~ sqrt(d-1) * polylog(2, 1/d)^(1/4) * d^(n - 1/2) * exp(2*sqrt(polylog(2, 1/d)*n)) / (2*sqrt(Pi)*n^(3/4)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A075900, A338673, A338674, A338675, A338676, A338677, A338678, A338679.

nmax = 30; CoefficientList[Series[Product[1/(1 - 3^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ polylog(2, 1/3)^(1/4) * 3^(n - 1/2) * exp(2*sqrt(polylog(2, 1/3)*n)) / (sqrt(2*Pi) * n^(3/4)), where polylog(2, 1/3) = 0.36621322997706348761674629...

a(n) = Sum_{k=0..n} p(n,k) * 3^(n-k), where p(n,k) is the number of partitions of n into k parts. - Ilya Gutkovskiy, Jun 08 2022

A338674 Expansion of Product_{k>=1} 1 / (1 - 5^(k-1)*x^k).

Original entry on oeis.org

1, 1, 6, 31, 181, 931, 5431, 27931, 159806, 834806, 4697306, 24478556, 137931681, 717306681, 3989650431, 20958791056, 115494337931, 604881056681, 3333662306681, 17439531447306, 95396181837931, 501716543166056, 2725636758009806, 14311071572462931, 77793648720900431

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246937, A300579, A338673, A338675, A338676, A338677, A338678, A338679.

nmax = 24; CoefficientList[Series[Product[1/(1 - 5^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 5^(n - k), {k, 0, n}], {n, 0, 24}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 5^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]

a(n) = Sum_{k=0..n} p(n,k) * 5^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ polylog(2, 1/5)^(1/4) * 5^(n - 1/2) * exp(2*sqrt(polylog(2, 1/5)*n)) / (sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338675 Expansion of Product_{k>=1} 1 / (1 - 6^(k-1)*x^k).

Original entry on oeis.org

1, 1, 7, 43, 295, 1807, 12391, 75895, 512647, 3179815, 21196807, 131258311, 875934727, 5416216711, 35763798535, 223059458311, 1461247179271, 9093600322567, 59586011601415, 370499158291975, 2411884242270727, 15072418547458567, 97530161503173127, 608700350537722375

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246938, A300579, A338673, A338674, A338676, A338677, A338678, A338679.

nmax = 23; CoefficientList[Series[Product[1/(1 - 6^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 6^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 6^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) = Sum_{k=0..n} p(n,k) * 6^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(5) * polylog(2, 1/6)^(1/4) * 6^(n - 1/2) * exp(2*sqrt(polylog(2, 1/6)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338676 Expansion of Product_{k>=1} 1 / (1 - 7^(k-1)*x^k).

Original entry on oeis.org

1, 1, 8, 57, 449, 3193, 25145, 178809, 1391314, 9996498, 76955586, 552257546, 4255024523, 30502987019, 232969386483, 1682476714724, 12762937304013, 92019035596293, 698222541789109, 5030814634614406, 37955614705675479, 274741644961416648, 2061916926761604144, 14909943849253537057

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246939, A300579, A338673, A338674, A338675, A338677, A338678, A338679.

nmax = 23; CoefficientList[Series[Product[1/(1 - 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 7^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) = Sum_{k=0..n} p(n,k) * 7^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(6) * polylog(2, 1/7)^(1/4) * 7^(n - 1/2) * exp(2*sqrt(polylog(2, 1/7)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338677 Expansion of Product_{k>=1} 1 / (1 - 8^(k-1)*x^k).

Original entry on oeis.org

1, 1, 9, 73, 649, 5257, 46729, 378505, 3331721, 27219593, 237491849, 1938544265, 16925054601, 138041874057, 1196384310921, 9820024329865, 84609648809609, 693596417152649, 5977550934234761, 48976660041553545, 419984680697190025, 3455551232025810569, 29494747047731910281

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246940, A300579, A338673, A338674, A338675, A338676, A338678, A338679.

nmax = 22; CoefficientList[Series[Product[1/(1 - 8^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 8^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

a(n) = Sum_{k=0..n} p(n,k) * 8^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(7) * polylog(2, 1/8)^(1/4) * 8^(n - 1/2) * exp(2*sqrt(polylog(2, 1/8)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338678 Expansion of Product_{k>=1} 1 / (1 - 9^(k-1)*x^k).

Original entry on oeis.org

1, 1, 10, 91, 901, 8191, 81091, 737191, 7239142, 66288142, 646149322, 5912729632, 57664985653, 527352541453, 5111015223223, 46998961540624, 453182267869615, 4163124744738505, 40151590267580785, 368699990679135946, 3540322181970716707, 32632895079429817528, 312061810101214595698

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246941, A300579, A338673, A338674, A338675, A338676, A338677, A338679.

nmax = 22; CoefficientList[Series[Product[1/(1 - 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 9^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

a(n) = Sum_{k=0..n} p(n,k) * 9^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(2) * polylog(2, 1/9)^(1/4) * 9^(n - 1/2) * exp(2*sqrt(polylog(2, 1/9)*n)) / (sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338679 Expansion of Product_{k>=1} 1 / (1 - 10^(k-1)*x^k).

Original entry on oeis.org

1, 1, 11, 111, 1211, 12211, 133211, 1343211, 14553211, 147653211, 1589753211, 16120753211, 173641753211, 1759951753211, 18855161753211, 192028261753211, 2048080361753211, 20841811361753211, 222333332361753211, 2261780642361753211, 24033895852361753211, 245331468952361753211

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

In general, if g.f. = Product_{k>=1} 1/(1 - d^(k-1)*x^k), where d > 1, then a(n) ~ sqrt(d-1) * polylog(2, 1/d)^(1/4) * d^(n - 1/2) * exp(2*sqrt(polylog(2, 1/d)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Cf. A008284, A075900, A246942, A300579, A338673, A338674, A338675, A338676, A338677, A338678.

nmax = 21; CoefficientList[Series[Product[1/(1 - 10^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 10^(n - k), {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 10^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

a(n) = Sum_{k=0..n} p(n,k) * 10^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ 3 * polylog(2, 1/10)^(1/4) *10^(n - 1/2) * exp(2*sqrt(polylog(2, 1/10)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344063 Expansion of Product_{k>=1} (1 + 4^(k-1)*x^k).

Original entry on oeis.org

1, 1, 4, 20, 80, 384, 1600, 7424, 30720, 143360, 593920, 2703360, 11403264, 51118080, 214958080, 965738496, 4047503360, 17951621120, 76168560640, 334202142720, 1411970498560, 6211596451840, 26203595472896, 114246130073600, 484815908372480, 2101441598586880, 8896148580335616

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A261568, A304961, A338673, A340103, A344062, A344064, A344065, A344066, A344067, A344068.

nmax = 26; CoefficientList[Series[Product[(1 + 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 4^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 26}]

seq(n)={Vec(prod(k=1, n, 1 + 4^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 4^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/4))^(1/4) * 4^n * exp(2*sqrt(-polylog(2, -1/4)*n)) / (2*sqrt(5*Pi/4)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

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

Original entry on oeis.org

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

A370434 Expansion of Product_{n>=1} (1 - 4^(n-1)x^n) (1 + 4^(n-1)*x^n)^2.

Original entry on oeis.org

1, 1, 3, 19, 60, 348, 1216, 6480, 23040, 121152, 445696, 2214912, 8475648, 40796160, 158564352, 754302976, 2949120000, 13694926848, 55180001280, 250151436288, 1008079994880, 4570684063744, 18552497111040, 82564035379200, 339344829186048, 1494986847682560, 6161930523770880

Offset: 0

Paul D. Hanna, Feb 26 2024

nonn

Compare to Product_{n>=1} (1 - 4^n*x^n) * (1 + 4^n*x^n)^2 = Sum_{n>=0} 4^(n*(n+1)/2) * x^(n*(n+1)/2).

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 60*x^4 + 348*x^5 + 1216*x^6 + 6480*x^7 + 23040*x^8 + 121152*x^9 + 445696*x^10 + 2214912*x^11 + 8475648*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 4*x^2)*(1 + 4*x^2)^2 * (1 - 16*x^3)*(1 + 16*x^3)^2 * (1 - 64*x^4)*(1 + 64*x^4)^2 * ... * (1 - 4^(n-1)*x^n)*(1 + 4^(n-1)*x^n)^2 * ...

Paul D. Hanna, Table of n, a(n) for n = 0..630

Cf. A370337, A370338, A370435, A344063, A338673.

{a(n) = polcoeff( prod(k=1,n, (1 - 4^(k-1)*x^k) * (1 + 4^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,40, print1(a(n),", "))

a(n) ~ c^(1/4) * 2^(2*n + 2) * exp(2*sqrt(c*n)) / (5 * sqrt(3*Pi) * n^(3/4)), where c = -2*polylog(2, -1/4) - polylog(2, 1/4). - Vaclav Kotesovec, Feb 27 2024

cp's OEIS Frontend

A300579 Expansion of Product_{k>=1} 1/(1 - 3^(k-1)*x^k).

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A338674 Expansion of Product_{k>=1} 1 / (1 - 5^(k-1)*x^k).

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A338675 Expansion of Product_{k>=1} 1 / (1 - 6^(k-1)*x^k).

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A338676 Expansion of Product_{k>=1} 1 / (1 - 7^(k-1)*x^k).

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A338677 Expansion of Product_{k>=1} 1 / (1 - 8^(k-1)*x^k).

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A338678 Expansion of Product_{k>=1} 1 / (1 - 9^(k-1)*x^k).

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A338679 Expansion of Product_{k>=1} 1 / (1 - 10^(k-1)*x^k).

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A344063 Expansion of Product_{k>=1} (1 + 4^(k-1)*x^k).

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

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A370434 Expansion of Product_{n>=1} (1 - 4^(n-1)*x^n) * (1 + 4^(n-1)*x^n)^2.

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A370434 Expansion of Product_{n>=1} (1 - 4^(n-1)x^n) (1 + 4^(n-1)*x^n)^2.