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.

A305886 Expansion of (1 - 1728*x)^(1/288).

Original entry on oeis.org

1, -6, -5166, -5940900, -7690495050, -10622111763060, -15285218827043340, -22626491069403298440, -34194284628635734767450, -52499624999832064779624900, -81615917024738927906404869540, -128166668244121840059567065130360
Offset: 0

Views

Author

Seiichi Manyama, Jun 13 2018

Keywords

Links

Crossrefs

Cf. A004996, A289366.

Programs

  • PARI
    N=20; x='x+O('x^N); Vec((1-1728*x)^(1/288))

Formula

a(n) = 6^n/n! * Product_{k=0..n-1} (288*k - 1) for n > 0.
a(n) = -1728^n * Gamma(n - 1/288) / (288 * Gamma(287/288) * Gamma(n+1)). - Vaclav Kotesovec, Jun 14 2018
a(n) ~ -2^(6*n - 5) * 3^(3*n - 2) / (Gamma(287/288) * n^(289/288)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +6*(-288*n+289)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

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

A248328 Square array read by antidiagonals downwards: super Patalan numbers of order 6.

Original entry on oeis.org

1, 6, 30, 126, 90, 990, 3276, 1260, 1980, 33660, 93366, 24570, 20790, 50490, 1161270, 2800980, 560196, 324324, 424116, 1393524, 40412196, 86830380, 14004900, 6162156, 5513508, 9754668, 40412196, 1414426860, 2753763480, 372130200, 132046200, 89791416, 108694872, 242473176, 1212365880
Offset: 0

Views

Author

Tom Richardson, Oct 04 2014

Keywords

Comments

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 6, A025751.

Examples

			T(0..4,0..4) is
  1          6         126       3276      93366
  30         90        1260      24570     560196
  990        1980      20790     324324    6162156
  33660      50490     424116    5513508   89791416
  1161270    1393524   9754668   108694872 1548901926

Links

Crossrefs

Cf. A068555, A025751, A004993 (first row), A004994 (first column), A004995 (second row), A004996 (second column), A248324, A248325, A248326, A248329, A248332.

Programs

  • PARI
    matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*36^(n+k)*binomial(n-1/6,n+k)) \\ Michel Marcus, Oct 09 2014

Formula

T(0,0)=1, T(n,k) = T(n-1,k)*(36*n-6)/(n+k), T(n,k) = T(n,k-1)*(36*k-30)/(n+k).
G.f.: (x/(1-36*x)^(5/6)+y/(1-36*y)^(1/6))/(x+y-36*x*y).
T(n,k) = (-1)^k*36^(n+k)*binomial(n-1/6,n+k).

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
Showing 1-5 of 5 results.

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