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 1-10 of 12 results. Next

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

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

A344064 Expansion of Product_{k>=1} (1 + 5^(k-1)*x^k).

Original entry on oeis.org

1, 1, 5, 30, 150, 875, 4500, 25625, 131250, 750000, 3843750, 21562500, 112109375, 621093750, 3222656250, 17880859375, 92578125000, 508300781250, 2658691406250, 14465332031250, 75439453125000, 411254882812500, 2142486572265625, 11590576171875000, 60722351074218750, 326728820800781250
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Crossrefs

Cf. A003056, A008289, A261569, A304961, A338674, A340103, A344062, A344063, A344065, A344066, A344067, A344068.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1 + 5^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 5^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 25}]
  • PARI
    seq(n)={Vec(prod(k=1, n, 1 + 5^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 5^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
a(n) ~ (-polylog(2, -1/5))^(1/4) * 5^n * exp(2*sqrt(-polylog(2, -1/5)*n)) / (2*sqrt(6*Pi/5)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344065 Expansion of Product_{k>=1} (1 + 6^(k-1)*x^k).

Original entry on oeis.org

1, 1, 6, 42, 252, 1728, 10584, 71280, 435456, 2939328, 17962560, 119532672, 739031040, 4867527168, 30051689472, 198147658752, 1221537687552, 7984437608448, 49643697954816, 321998350270464, 1997815999463424, 12977575759282176, 80455233450737664, 519208351807832064
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Crossrefs

Cf. A003056, A008289, A304961, A338675, A340103, A344062, A344063, A344064, A344066, A344067, A344068.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + 6^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 6^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 23}]
  • PARI
    seq(n)={Vec(prod(k=1, n, 1 + 6^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 6^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
a(n) ~ (-polylog(2, -1/6))^(1/4) * 6^n * exp(2*sqrt(-polylog(2, -1/6)*n)) / (2*sqrt(7*Pi/6)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344066 Expansion of Product_{k>=1} (1 + 7^(k-1)*x^k).

Original entry on oeis.org

1, 1, 7, 56, 392, 3087, 21952, 170471, 1210104, 9411920, 66824632, 513890832, 3683707839, 28086110472, 201122377288, 1534688027817, 10978118077136, 83158453503608, 599161640356888, 4508826988300152, 32435340235930576, 244366486039786096, 1756858874561956865, 13161303959340223232
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Crossrefs

Cf. A003056, A008289, A304961, A338676, A340103, A344062, A344063, A344064, A344065, A344067, A344068.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 7^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 23}]
  • PARI
    seq(n)={Vec(prod(k=1, n, 1 + 7^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 7^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
a(n) ~ (-polylog(2, -1/7))^(1/4) * 7^n * exp(2*sqrt(-polylog(2, -1/7)*n)) / (4*sqrt(2*Pi/7)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344067 Expansion of Product_{k>=1} (1 + 8^(k-1)*x^k).

Original entry on oeis.org

1, 1, 8, 72, 576, 5120, 41472, 364544, 2949120, 25952256, 209977344, 1830813696, 14931722240, 129251672064, 1053340729344, 9123584278528, 74294344286208, 639503450505216, 5239722662166528, 44846880273727488, 367008185258606592, 3144110674230116352, 25718087147075928064
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Crossrefs

Cf. A003056, A008289, A304961, A338677, A340103, A344062, A344063, A344064, A344065, A344066, A344068.

Programs

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

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 8^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
a(n) ~ (-polylog(2, -1/8))^(1/4) * 8^n * exp(2*sqrt(-polylog(2, -1/8)*n)) / (6*sqrt(Pi/8)*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

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, -1, -2, -7, -11, -43, -65, -259, -146, -1798, 826, -8116, 17593, -35089, 301903, -308464, 3582403, 157367, 28816009, 9388694, 329375419, -61352008, 2991009094, 509592773, 23675224255, 1207374806, 229200996508, -129896994130, 2090952547882, -816324790165, 14079091274800
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2022

Keywords

Crossrefs

Cf. A032308, A242587, A261582, A292128, A300579, A344062, A352402, A352786.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 3^(n - k), {k, 0, n}], {n, 0, 30}]

Formula

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 3^(n-k), where p(n,k) is the number of partitions of n into k parts.

A370338 Expansion of Product_{n>=1} (1 - 3^(n-1)*x^n) * (1 + 3^(n-1)*x^n)^2.

Original entry on oeis.org

1, 1, 2, 11, 24, 114, 297, 1224, 3240, 13230, 37017, 138510, 407754, 1469664, 4413366, 15717969, 47239200, 163408266, 511758000, 1719152586, 5348422224, 18083342907, 56672868240, 187301066040, 594207370746, 1947548449296, 6185182455792, 20263641256656, 64084643627283
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2024

Keywords

Comments

Compare to Product_{n>=1} (1 - 3^n*x^n) * (1 + 3^n*x^n)^2 = Sum_{n>=0} 3^(n*(n+1)/2) * x^(n*(n+1)/2).
In general, for d > 1, if g.f. = Product_{k>=1} (1 - d^(k-1)*x^k) * (1 + d^(k-1)*x^k)^2, then a(n) ~ c^(1/4) * d^(n + 3/2) * exp(2*sqrt(c*n)) / (2 * sqrt((d-1)*Pi) * (d+1) * n^(3/4)), where c = -2*polylog(2, -1/d) - polylog(2, 1/d). - Vaclav Kotesovec, Feb 26 2024

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 24*x^4 + 114*x^5 + 297*x^6 + 1224*x^7 + 3240*x^8 + 13230*x^9 + 37017*x^10 + 138510*x^11 + 407754*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 3*x^2)*(1 + 3*x^2)^2 * (1 - 9*x^3)*(1 + 9*x^3)^2 * (1 - 27*x^4)*(1 + 27*x^4)^2 * ... * (1 - 3^(n-1)*x^n)*(1 + 3^(n-1)*x^n)^2 * ...
Compare A(x) to the series that results from a similar infinite product:
(1 - 3*x)*(1 + 3*x)^2 * (1 - 9*x^2)*(1 + 9*x^2)^2 * (1 - 27*x^3)*(1 + 27*x^3)^2 * (1 - 81*x^4)*(1 + 81*x^4)^2 * ... = 1 + 3*x + 27*x^3 + 729*x^6 + 59049*x^10 + 14348907*x^15 + 10460353203*x^21 + 22876792454961*x^28 + ...

Links

Crossrefs

Cf. A300579, A344062, A370337.

Programs

  • PARI
    {a(n) = polcoeff( prod(k=1,n, (1 - 3^(k-1)*x^k) * (1 + 3^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,30, print1(a(n),", "))

Formula

a(n) ~ c^(1/4) * 3^(n + 3/2) * exp(2*sqrt(c*n)) / (2^(7/2) * sqrt(Pi) * n^(3/4)), where c = -2*polylog(2,-1/3) - polylog(2,1/3) = 0.2518530229985534570173197... - Vaclav Kotesovec, Feb 26 2024

A370711 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/2).

Original entry on oeis.org

1, 6, 6, 348, -570, 12084, -31332, 780792, -6111930, 65506884, -599418444, 6707736456, -69508986852, 738378468744, -7878832564872, 85524000547056, -929068361832378, 10158667075255524, -111690827626777788, 1234592278534799592, -13700571880245603276, 152613494540593338264
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

Crossrefs

Cf. A032308, A370714.
Cf. A300579, A344062.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(4*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-3, x]/4], {x, 0, nmax}], x] * 4^Range[0, nmax]

Formula

G.f.: Product_{k>=1} (1 + 3*(4*x)^k)^(1/2).
a(n) ~ (-1)^(n+1) * c * 12^n / n^(3/2), where c = QPochhammer(-1/3)^(1/2) / (2*sqrt(Pi)) = 0.311283382185276347775502154581850436407169685238...

A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

Original entry on oeis.org

1, 3, 0, 99, -270, 2430, -10287, 105462, -750141, 5702481, -42623901, 347424633, -2779077762, 22353287634, -181730796723, 1493711042589, -12321529794261, 102125312638713, -850797139405887, 7120067746384863, -59800770201017934, 503922807927384129, -4259721779079782751
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

Crossrefs

Cf. A032308, A370710.
Cf. A300579, A344062.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(QPochhammer[-3, x]/4)^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

Formula

G.f.: Product_{k>=1} (1 + 3*(3*x)^k)^(1/3).
a(n) ~ (-1)^(n+1) * c * 9^n / n^(4/3), where c = QPochhammer(-1/3)^(1/3) / (3*Gamma(2/3)) = 0.26286302373105271371291957730496322329245126572...
Showing 1-10 of 12 results. Next

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