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.

Previous Showing 11-14 of 14 results.

A111995 Seventh convolution of Schroeder's (second problem) numbers A001003(n), n >= 0.

Original entry on oeis.org

1, 7, 42, 238, 1316, 7196, 39158, 212738, 1155889, 6287015, 34249404, 186920468, 1022134288, 5600420336, 30745867316, 169116129308, 931937277257, 5144687596447, 28449040406262, 157571572143538, 874089046798212
Offset: 0

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Author

Wolfdieter Lang, Sep 12 2005

Keywords

Links

Crossrefs

Cf. Seventh column of convolution triangle A011117.

Programs

  • Mathematica
    CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
  • PARI
    my(x='x+O('x^50)); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^7) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^7.
a(n) = (7/n)*Sum_{k=1..n} binomial(n,k)*binomial(n+k+6,k-1).
a(n) = 7*hypergeom([1-n, n+8], [2], -1), n >= 1, a(0)=1.
a(n) = fourth binomial transform of 1,3,2,6,4,12. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
Recurrence: n*(n+7)*a(n) = (7*n^2+37*n+12)*a(n-1) - (7*n^2+19*n-24)*a(n-2) + (n-3)*(n+4)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 7*sqrt(3*sqrt(2)-4)*(99-70*sqrt(2)) * (3+2*sqrt(2))^(n+7)/(32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

Extensions

Incorrect formula removed by Jason Yuen, Sep 07 2024

A111996 Eighth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

Original entry on oeis.org

1, 8, 52, 312, 1802, 10200, 57092, 317544, 1760035, 9738160, 53844184, 297717712, 1646893140, 9116815952, 50514367512, 280173703472, 1555632093093, 8647009926904, 48117998453036, 268057662257096, 1494927614877214
Offset: 0

Views

Author

Wolfdieter Lang, Sep 12 2005

Keywords

Links

Crossrefs

Cf. Eighth column of convolution triangle A011117.

Programs

  • Mathematica
    CoefficientList[Series[((1+x-Sqrt[1-6x+x^2])/(4x))^8, {x,0,20}], x]  (* Harvey P. Dale, Apr 01 2011 *)
  • PARI
    x='x+O(x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^8) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^8.
a(n)= (8/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+7,k-1).
a(n) = 8*hypergeom([1-n, n+9], [2], -1), n>=1, a(0)=1.
Recurrence: n*(n+8)*a(n) = (7*n^2+44*n+21)*a(n-1) - (7*n^2+26*n-24)*a(n-2) + (n-3)*(n+5)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ sqrt(3*sqrt(2)-4)*(338-239*sqrt(2)) * (3+2*sqrt(2))^(n+8)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

A111997 Ninth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

Original entry on oeis.org

1, 9, 63, 399, 2403, 14067, 80949, 460845, 2605590, 14666470, 82320714, 461238282, 2581644378, 14442658074, 80785970838, 451934259654, 2528977211775, 14157983986839, 79302044283297, 444448115168049, 2492468172937125
Offset: 0

Views

Author

Wolfdieter Lang, Sep 12 2005

Keywords

Links

Crossrefs

Ninth column of convolution triangle A011117.

Programs

  • Mathematica
    CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^9, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
  • PARI
    x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^9) \\ G. C. Greubel, Mar 17 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^9.
a(n) = (9/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+8,k-1).
a(n) = 9*hypergeom([1-n, n+10], [2], -1), n>=1, a(0)=1.
Recurrence: n*(n+9)*a(n) = (7*n^2+51*n+32)*a(n-1) - (7*n^2+33*n-22)*a(n-2) + (n-3)*(n+6)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 9*sqrt(3*sqrt(2)-4)*(577-408*sqrt(2)) * (3+2*sqrt(2))^(n+9)/(64*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

A111998 Tenth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

Original entry on oeis.org

1, 10, 75, 500, 3135, 18962, 112125, 653200, 3766950, 21571500, 122920642, 697994760, 3953743250, 22357130700, 126273263510, 712639689168, 4019975635855, 22671014908550, 127846248597125, 720994336613980
Offset: 0

Views

Author

Wolfdieter Lang, Sep 12 2005

Keywords

Links

Crossrefs

Cf. Tenth column of convolution triangle A011117.

Programs

  • Mathematica
    CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^10, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
  • PARI
    x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^10) \\ G. C. Greubel, Mar 17 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^10.
a(n) = (10/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+9,k-1).
a(n) = 10*hypergeom([1-n, n+11], [2], -1), n>=1, a(0)=1.
Contribution from Vaclav Kotesovec, Oct 18 2012: (Start)
Recurrence: n*(n+10)*a(n) = (7*n^2+58*n+45)*a(n-1) - (7*n^2+40*n-18)*a(n-2) + (n-3)*(n+7)*a(n-3)
a(n) ~ 5*sqrt(3*sqrt(2)-4)*(1970-1393*sqrt(2)) * (3+2*sqrt(2))^(n+10)/(64*sqrt(Pi)*n^(3/2))
Generally, G.f. = ((1+x-sqrt(1-6*x+x^2))/(4*x))^k is asymptotic to a(n) ~ sqrt(3*sqrt(2)-4)*k*(1-1/sqrt(2))^(k-1) * (3+2*sqrt(2))^(n+k)/(4*sqrt(Pi)*n^(3/2)).
(End)
Previous Showing 11-14 of 14 results.

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