cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 09 2018

Keywords

Comments

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)).

Links

Crossrefs

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

Programs

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

Formula

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

A338673 Expansion of Product_{k>=1} 1 / (1 - 4^(k-1)*x^k).

Original entry on oeis.org

1, 1, 5, 21, 101, 421, 2021, 8421, 39397, 167397, 766437, 3244517, 14881253, 62804453, 283415013, 1210159589, 5401907685, 22966866405, 102497423845, 435085808101, 1925197238757, 8215432696293, 36068400468453, 153579729097189, 674546796630501, 2866238341681637, 12508012102193637
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

Cf. A008284, A075900, A246936, A300579, A338674, A338675, A338676, A338677, A338678, A338679.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Product[1/(1 - 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 4^(n - k), {k, 0, n}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 4^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ sqrt(3) * polylog(2, 1/4)^(1/4) * 4^(n - 1/2) * exp(2*sqrt(polylog(2, 1/4)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

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

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

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

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

A344068 Expansion of Product_{k>=1} (1 + 9^(k-1)*x^k).

Original entry on oeis.org

1, 1, 9, 90, 810, 8019, 72900, 715149, 6495390, 63772920, 579270690, 5643903420, 51613018479, 499772430810, 4567687565310, 44250780833091, 404188047763920, 3894703308072990, 35764052204589030, 342923118899865390, 3146016498406236720, 30187757787717436380, 276843069234653897241
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Comments

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

Crossrefs

Cf. A003056, A008289, A304961, A338678, A340103, A344062, A344063, A344064, A344065, A344066, A344067.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 9^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 22}]
  • PARI
    seq(n)={Vec(prod(k=1, n, 1 + 9^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 9^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
a(n) ~ (-polylog(2, -1/9))^(1/4) * 9^n * exp(2*sqrt(-polylog(2, -1/9)*n)) / (2*sqrt(10*Pi/9)*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

Views

Author

Ilya Gutkovskiy, Apr 24 2021

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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
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