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.

A303135 Expansion of Product_{n>=1} (1 - (16*x)^n)^(-1/4).

Original entry on oeis.org

1, 4, 104, 1760, 39520, 590720, 14285056, 205151232, 4596467200, 75375073280, 1504196046848, 23673049726976, 525315968712704, 7912159583600640, 158055039529779200, 2726833423421800448, 51889395654107463680, 840470097284214292480, 16765991910040314839040
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): A000041 (b=1), A271235 (b=2), A271236 (b=3), this sequence (b=4), A303136 (b=5).
Cf. A303131, A303153.

Programs

  • Mathematica
    CoefficientList[Series[1/QPochhammer[16*x]^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2018 *)

Formula

a(n) ~ exp(sqrt(n/6)*Pi) * 2^(4*n - 33/16) / (3^(5/16) * n^(13/16)). - Vaclav Kotesovec, Apr 19 2018

A303132 Expansion of Product_{n>=1} (1 + (25*x)^n)^(-1/5).

Original entry on oeis.org

1, -5, -50, -3875, 2500, -2046250, -12409375, -1087687500, 13232343750, -907225000000, 1545669140625, -362705679687500, 6007095839843750, -224713698632812500, 2118331116210937500, -226812683210205078125, 4765872641563720703125
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.
In general, for h>=1, if g.f. = Product_{k>=1} (1 + (h^2*x)^k)^(-1/h), then a(n) ~ (-1)^n * exp(Pi*sqrt(n/(6*h))) * h^(2*n) / (2^(7/4) * 3^(1/4) * h^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

Links

Crossrefs

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), A303130 (b=3), A303131 (b=4), this sequence (b=5).
Cf. A303125, A303136.

Programs

  • Mathematica
    CoefficientList[Series[(2/QPochhammer[-1, 25*x])^(1/5), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(n/30)) * 5^(2*n - 1/4) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A303124 Expansion of Product_{n>=1} (1 + (16*x)^n)^(1/4).

Original entry on oeis.org

1, 4, 40, 1504, 10336, 387968, 5349632, 111442944, 1100563968, 36711258112, 493805416448, 9186633203712, 134635599806464, 2648342619422720, 43443234834350080, 938422838970810368, 11378951438668791808, 224791017150689574912, 4129154423023897411584
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): A000009 (b=1), A298994 (b=2), A303074 (b=3), this sequence (b=4), A303125 (b=5).
Cf. A303131, A303153.

Programs

  • Mathematica
    CoefficientList[Series[(QPochhammer[-1, 16*x]/2)^(1/4), {x, 0, 20}],
x] (* Vaclav Kotesovec, Apr 19 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+(16*x)^k)^(1/4)))

Formula

a(n) ~ 2^(4*n - 17/8) * exp(sqrt(n/3)*Pi/2) / (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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