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 1-5 of 5 results.

A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

Original entry on oeis.org

1, -2, -10, -20, -90, 132, -836, 6040, 2310, 60180, 180308, 1662568, -2995620, 24401320, 44072120, -102437328, 19390406, 2649221300, -10584460060, 14475802440, -228570333836, -815899620616, 2088529753800, -5590702681520, -100828534100580, -172013432412024
Offset: 0

Views

Author

William J. Keith, Jan 18 2018

Keywords

Comments

The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution.
Sequence appears to be positive and negative roughly half the time.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Links

Crossrefs

Cf. A000108, A271235, A298994.
Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), this sequence (b=2), A303152 (b=3), A303153 (b=4), A303154 (b=5).

Programs

  • Mathematica
    Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}]
  • PARI
    q='q+O('q^99); Vec(eta(4*q)^(1/2)) \\ Altug Alkan, Apr 20 2018

Formula

G.f.: Product_{k>=1} (1 - (4x)^k)^(1/2).

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

A303153 Expansion of Product_{n>=1} (1 - (16*x)^n)^(1/4).

Original entry on oeis.org

1, -4, -88, -992, -19360, -97152, -4296448, 4539392, -568015360, -127621120, -39357927424, 2424998313984, -38804685471744, 799759166930944, 4879962868940800, 41563181340426240, 585185165832486912, 55834295603426754560, -75535223925056208896
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): A010815 (b=1), A298411 (b=2), A303152 (b=3), this sequence (b=4), A303154 (b=5).
Cf. A303124, A303135.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(16*x)^k)^(1/4)))

A303152 Expansion of Product_{n>=1} (1 - (9*x)^n)^(1/3).

Original entry on oeis.org

1, -3, -36, -207, -2214, -2754, -138591, 547722, -3730293, 30138075, 133709535, 7735237479, -35284817430, 702841889322, 3056530613769, 9493893988155, 112554319443867, 3822223052352735, -3940051663965051, 250298859930263181, -551418001934739786, 1061747224529191191
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): A010815 (b=1), A298411 (b=2), this sequence (b=3), A303153 (b=4), A303154 (b=5).
Cf. A271236, A303074.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(9*x)^k)^(1/3)))

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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.