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.

A301271 Expansion of (1-16*x)^(1/8).

Original entry on oeis.org

1, -2, -14, -140, -1610, -19964, -259532, -3485144, -47920730, -670890220, -9526641124, -136837208872, -1984139528644, -28998962341720, -426699017313880, -6315145456245424, -93937788661650682, -1403541077650545484, -21053116164758182260, -316904801216886322440
Offset: 0

Views

Author

Seiichi Manyama, Jun 15 2018

Keywords

Links

Crossrefs

Cf. A003557, A007947.
(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), this sequence (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

Programs

  • PARI
    N=20; x='x+O('x^N); Vec((1-16*x)^(1/8))

Formula

a(n) = 2^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = -sqrt(2-sqrt(2)) * Gamma(1/8) * Gamma(n-1/8) * 16^(n-1) / (Pi*Gamma(n+1)). - Vaclav Kotesovec, Jun 16 2018
a(n) ~ -2^(4*n-3) / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +2*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = -2*A097184(n-1). - R. J. Mathar, Jan 20 2020

A303007 Expansion of (1-240*x)^(1/8).

Original entry on oeis.org

1, -30, -3150, -472500, -81506250, -15160162500, -2956231687500, -595469525625000, -122815589660156250, -25791273828632812500, -5493541325498789062500, -1183608449221102734375000, -257434837705589844726562500, -56437637496994696728515625000
Offset: 0

Views

Author

Seiichi Manyama, Jun 15 2018

Keywords

Links

Crossrefs

Cf. A003557, A007947, A049210, A108091.
(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), A301271 (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), this sequence (b=240), A303055 (b=504), A305886 (b=1728).

Programs

  • Mathematica
    CoefficientList[Series[Surd[1-240x,8],{x,0,20}],x] (* Harvey P. Dale, Aug 29 2024 *)
  • PARI
    N=20; x='x+O('x^N); Vec((1-240*x)^(1/8))

Formula

a(n) = 30^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = 15^n * A301271(n).
a(n) ~ -2^(4*n - 3) * 15^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +30*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A305991 Expansion of (1-27*x)^(1/9).

Original entry on oeis.org

1, -3, -36, -612, -11934, -250614, -5513508, -125235396, -2911722957, -68910776649, -1653858639576, -40143659706072, -983519662798764, -24285370135261788, -603664914790793016, -15091622869769825400, -379177024602966863175, -9568643738510163782475
Offset: 0

Views

Author

Seiichi Manyama, Jun 16 2018

Keywords

Links

Crossrefs

Cf. A003557, A007947.
(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), A301271 (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), this sequence (b=27), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

Programs

  • PARI
    N=20; x='x+O('x^N); Vec((1-27*x)^(1/9))

Formula

a(n) = 3^n/n! * Product_{k=0..n-1} (9*k - 1) for n > 0.
a(n) ~ 27^n / (Gamma(-1/9) * n^(10/9)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +3*(-9*n+10)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

A108734 Expansion of (1 + 24*x)^(1/2).

Original entry on oeis.org

1, 12, -72, 864, -12960, 217728, -3919104, 73903104, -1441110528, 28822210560, -587973095424, 12187078705152, -255928652808192, 5433562167312384, -116433475013836800, 2514963060298874880, -54700446561500528640, 1196974477698717450240, -26333438509371783905280, 582107588101902591590400
Offset: 0

Views

Author

N. J. A. Sloane, Jun 22 2005

Keywords

Crossrefs

Cf. A108735.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 24 x)^(1/2), {x, 0, 20}], x] (* Vincenzo Librandi, Jan 21 2020 *)

Formula

a(n) = 2^n*A108735(n). - Zak Seidov, Jun 24 2005
D-finite with recurrence: n*a(n) +12*(2*n-3)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
From Amiram Eldar, May 28 2022: (Start)
Sum_{n>=0} 1/a(n) = (672 - 72*arcsinh(1/(2*sqrt(6)))/5)/625.
Sum_{n>=0} 1/a(n) = (480 - 72*arcsin(1/(2*sqrt(6)))/sqrt(23))/529. (End)
Showing 1-4 of 4 results.

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

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