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.

A143168 Third column of triangle A000369: |S2(-3; n+3, 3)|.

Original entry on oeis.org

1, 18, 345, 7650, 196245, 5755050, 190482705, 7034400450, 286988226525, 12826061498250, 623403611055225, 32747785180560450, 1849179329801929125, 111713055889014830250, 7190273917194645902625, 491244630824362410245250, 35508203161436371983742125
Offset: 0

Views

Author

Wolfdieter Lang, Sep 15 2008

Keywords

Crossrefs

Second column of A000369 is A143167, fourth one is A143169.

Programs

  • PARI
    lista(nn) = default(seriesprecision, nn); Vec(serlaplace(deriv(deriv(deriv(((1-(1-4*x)^(1/4))^3)/3!))))); \\ Michel Marcus, Dec 24 2017

Formula

a(n) = A000369(n+3,3) = |S2(-3; n+3, 3)|, n >= 0.
E.g.f.: (d^3/dx^3)((1-(1-4*x)^(1/4))^3)/3! = (-1/2)*(20*x - 5 - 21*(1-4*x)^(1/2)+24*(1-4*x)^(3/4))/(1-4*x)^(13/4).

A143170 Fifth column of triangle A000369: |S2(-3;n+5,5)|.

Original entry on oeis.org

1, 45, 1680, 62790, 2471175, 104085135, 4712781150, 229345716600, 11970744110325, 668241679730625, 39773331191493900, 2516317288024790250, 168723807382851595875, 11956978164372003637875, 893260022082269487896250, 70178395183380972653665500
Offset: 0

Views

Author

Wolfdieter Lang, Sep 15 2008

Keywords

Crossrefs

Fourth column of A000369 is A143169.

Formula

a(n) = A000369(n+5,5) = |S2(-3;n+5,5)|, n >= 0.
E.g.f.: d^5/dx^5 ((1-(1-4*x)^(1/4))^5)/5! = (1/4)*(539+154*x+585*(1-4*x)^(1/2)-1120*(1-4*x)^(1/4))/(1-4*x)^(19/4).

A145562 Second column (m=2) of triangle A049029 (S2(5)).

Original entry on oeis.org

1, 15, 255, 5175, 123795, 3427515, 108046575, 3824996175, 150346471275, 6499426608675, 306553491419175, 15668768604864375, 862827112324051875, 50929793720847916875, 3208139019437586609375, 214817175616326677769375, 15237402314816854944646875
Offset: 0

Views

Author

Wolfdieter Lang, Oct 17 2008, Dec 04 2008

Keywords

Links

Crossrefs

First column: A007696 (4-factorials). Third column A143169.

Programs

  • Mathematica
    FullSimplify@Table[(-4)^n(8Sqrt[Pi]/Gamma[-3/2-n]-5Gamma[-5/4]/Gamma[-5/4-n]),{n, 0, 20}] (* Benedict W. J. Irwin, Apr 06 2017 *)
  • PARI
    x='x+O('x^50); Vec(serlaplace((6 - 5*(1-4*x)^(1/4))/(1 -4*x)^(5/2))) \\ G. C. Greubel, May 25 2017

Formula

a(n) = A049029(n+2,2),n>=0.
E.g.f. with offset n=2: ((-1+(1-4*x)^(-1/4))^2)/2!.
E.g.f.: (6 - 5*(1-4*x)^(1/4))/(1-4*x)^(5/2) (offset n=0).
a(n) = (-4)^n*(8*Sqrt(Pi)/Gamma(-3/2-n)-5*Gamma(-5/4)/Gamma(-5/4-n)). - Benedict W. J. Irwin, Apr 06 2017

A383196 Expansion of e.g.f. (1/(1 - 3*x)^(1/3) - 1)^3 / 6.

Original entry on oeis.org

0, 0, 0, 1, 24, 520, 11880, 295960, 8090880, 242280640, 7912262400, 280384720000, 10727852889600, 441104638374400, 19407654326860800, 910140650683264000, 45332366929833984000, 2390437704451084288000, 133060566042200788992000, 7797805996570952986624000
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2025

Keywords

Crossrefs

Column k=3 of A371080.
Cf. A001754, A035119, A143169.

Programs

  • PARI
    a(n) = sum(k=3, n, 3^(n-k)*abs(stirling(n, k, 1))*stirling(k, 3, 2));

Formula

a(n) = Sum_{k=3..n} 3^(n-k) * |Stirling1(n,k)| * Stirling2(k,3).
a(n) ~ sqrt(Pi/2) * n^(n + 1/2) * 3^(n-1) * exp(-n) * (1 - 3/(Gamma(2/3)*n^(1/3))). - Vaclav Kotesovec, May 03 2025
Showing 1-4 of 4 results.

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