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 1-4 of 4 results.

A303136 Expansion of Product_{n>=1} (1 - (25*x)^n)^(-1/5).

Original entry on oeis.org

1, 5, 200, 5125, 177500, 3952500, 150715625, 3185187500, 112844843750, 2783033593750, 86330708203125, 2019237027343750, 72195817812500000, 1591910699609375000, 50158322275878906250, 1322261581989501953125, 39183430287559814453125, 946961406814801025390625
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/5, g(n) = 25^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), A271236 (b=3), A303135 (b=4), this sequence (b=5).
Cf. A303132, A303154.

Programs

  • Mathematica
    CoefficientList[Series[1/QPochhammer[25*x]^(1/5), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2018 *)
CoefficientList[Series[Product[(1-(25x)^n)^(-1/5),{n,20}],{x,0,20}],x] (* Harvey P. Dale, Nov 04 2021 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/15)) * 5^(2*n - 3/10) / (2^(7/5) * 3^(3/10) * n^(4/5)). - Vaclav Kotesovec, Apr 19 2018

A303131 Expansion of Product_{n>=1} (1 + (16*x)^n)^(-1/4).

Original entry on oeis.org

1, -4, -24, -1248, 1632, -267136, -669440, -56925184, 597165568, -19934894080, 61831327744, -3209599664128, 47593545383936, -840449808072704, 8113679782510592, -350055154021040128, 5703847053344768000, -57129722970675609600, 704939718429511778304
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/4, g(n) = -16^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), A303130 (b=3), this sequence (b=4), A303132 (b=5).
Cf. A303124, A303135.

Programs

  • Mathematica
    CoefficientList[Series[(2/QPochhammer[-1, 16*x])^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(n/24)) * 2^(4*n - 9/4) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A303125 Expansion of Product_{n>=1} (1 + (25*x)^n)^(1/5).

Original entry on oeis.org

1, 5, 75, 4500, 43125, 2765000, 55871875, 1876671875, 25128437500, 1495793359375, 28953471875000, 871257974609375, 18280647500000000, 596362168603515625, 14502797130615234375, 519397373566650390625, 8604439235863037109375
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/5, g(n) = -25^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(1/b): A000009 (b=1), A298994 (b=2), A303074 (b=3), A303124 (b=4), this sequence (b=5).
Cf. A303132, A303154.

Programs

  • Mathematica
    CoefficientList[Series[(QPochhammer[-1, 25*x]/2)^(1/5), {x, 0, 20}],
x] (* Vaclav Kotesovec, Apr 19 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+(25*x)^k)^(1/5)))

Formula

a(n) ~ 5^(2*n - 1/4) * exp(Pi*sqrt(n/15)) / (2^(8/5) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 19 2018

A303130 Expansion of Product_{n>=1} (1 + (9*x)^n)^(-1/3).

Original entry on oeis.org

1, -3, -9, -288, 459, -19278, -1539, -1265301, 10734525, -147277926, 520204923, -7511358663, 88687160577, -668191863951, 5357547144702, -87542760890124, 967961569696722, -5115624735401361, 46065749188891275, -430898393089547667, 6203508335817169257
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = -9^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), this sequence (b=3), A303131 (b=4), A303132 (b=5).
Cf. A303074.

Programs

  • Mathematica
    CoefficientList[Series[(2/QPochhammer[-1, 9*x])^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, (1 + (9*x)^k)^(-1/3))) \\ Altug Alkan, Apr 20 2018

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(n/18)) * 3^(2*n - 1/2) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018
Showing 1-4 of 4 results.

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