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.

A344062 Expansion of Product_{k>=1} (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 12, 36, 135, 432, 1539, 4860, 17496, 55404, 192456, 623295, 2125764, 6849684, 23442453, 75110328, 252965916, 822670668, 2735858268, 8838926712, 29501352792, 95090206689, 314068876416, 1018141045092, 3342663979092, 10798571289897, 35481518064576
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Crossrefs

Cf. A003056, A008289, A032308, A300579, A304961, A340103, A344063, A344064, A344065, A344066, A344067, A344068.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Product[(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 27}]
  • PARI
    seq(n)={Vec(prod(k=1, n, 1 + 3^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

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

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

A344094 a(n) = [x^n] Product_{k>=1} (1 + n^(k+1)*x^k).

Original entry on oeis.org

1, 1, 8, 324, 5120, 171875, 13716864, 409300871, 20535312384, 976299632280, 144100000000000, 6251749326428232, 484144254340300800, 31585633366079696358, 2452531026468909711360, 483966896057281494140625, 31314307295813796764844032, 3176091371161687418319418614
Offset: 0

Views

Author

Vaclav Kotesovec, May 09 2021

Keywords

Links

Crossrefs

Cf. A292305, A340103, A344095.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1+n^(k+1)*x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[SeriesCoefficient[QPochhammer[-n, n*x]/(n+1), {x, 0, n}], {n, 1, 20}]]

A346452 a(n) = [x^n] Product_{k>=1} (1 + n^(k-1)*x^k) / (1 - n^(k-1)*x^k).

Original entry on oeis.org

1, 2, 6, 32, 242, 2472, 31850, 490912, 8897058, 185373722, 4344896842, 113667973344, 3283155709106, 103486639506824, 3544144942010010, 131038318999642112, 5192873357864667202, 219952910549005233840, 9916266932124308092586, 473513072853351852164160
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 19 2021

Keywords

Links

Crossrefs

Cf. A340103, A338697, A346454.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + n^(k - 1)*x^k)/(1 - n^(k - 1)*x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[SeriesCoefficient[2*x + (n - 1)/(n + 1)*QPochhammer[-1/n, n*x]/QPochhammer[1/n, n*x], {x, 0, n}], {n, 1, 20}]]

Formula

a(n) ~ c * n^(n-1), where c = sqrt(2) * BesselI(1, 2*sqrt(2)) = 4.78966619...
Showing 1-9 of 9 results.

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