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 1-2 of 2 results.

A305032 a(0) = 0, a(1) = 1 and a(n) = 6*a(n-1)/(n-1) + 4*a(n-2) for n > 1.

Original entry on oeis.org

0, 1, 6, 22, 68, 190, 500, 1260, 3080, 7350, 17220, 39732, 90552, 204204, 456456, 1012440, 2230800, 4886310, 10647780, 23094500, 49884120, 107343236, 230205976, 492156392, 1049212528, 2230928700, 4732273000, 10015777800, 21154820400, 44596287000, 93846099600
Offset: 0

Views

Author

Seiichi Manyama, May 24 2018

Keywords

Comments

Let a(0) = 0, a(1) = 1 and a(n) = 2*m*a(n-1)/(n-1) + k^2*a(n-2), for n > 1, then the g.f. is x/(2*m) * d/dx ((1 + k*x)/(1 - k*x))^(m/k).

Links

Crossrefs

Cf. A100071, A304944, A305031.

Programs

  • Magma
    [n le 2 select n-1 else 2*(3*Self(n-1) + 2*(n-2)*Self(n-2))/(n-2): n in [1..40]]; // G. C. Greubel, Jun 07 2023
  • Mathematica
    CoefficientList[Series[x*Sqrt[1-4*x^2]/(1-2*x)^3, {x,0,40}], x] (* G. C. Greubel, Jun 07 2023 *)
  • SageMath
    @CachedFunction
def a(n): # b = A305032
    if n<2: return n
    else: return 2*(3*a(n-1) + 2*(n-1)*a(n-2))//(n-1)
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

a(n) = n*A305031(n)/6.
G.f.: x*sqrt(1-4*x^2)/(1-2*x)^3.

A305612 Expansion of 1/2 * (((1 + 2*x)/(1 - 2*x))^(3/2) - 1).

Original entry on oeis.org

0, 3, 9, 22, 51, 114, 250, 540, 1155, 2450, 5166, 10836, 22638, 47124, 97812, 202488, 418275, 862290, 1774630, 3646500, 7482618, 15334748, 31391724, 64194312, 131151566, 267711444, 546031500, 1112864200, 2266587900, 4613409000, 9384609960, 19079454960
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2018

Keywords

Comments

Let 1/2 * (((1 + k*x)/(1 - k*x))^(m/k) - 1) = a(0) + a(1)*x + a(2)*x^2 + ...
Then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Links

Crossrefs

1/2 * (((1 + 2*x)/(1 - 2*x))^(m/2) - 1): A001405(n-1) (m=1), this sequence (m=3).
Cf. A305031.

Programs

  • Maple
    seq(coeff(series((1/2)*(((1+2*x)/(1-2*x))^(3/2)-1), x,n+1),x,n),n=0..35); # Muniru A Asiru, Jun 06 2018
  • Mathematica
    CoefficientList[Series[((((1+2x)/(1-2x))^(3/2))-1)/2,{x,0,40}],x] (* Harvey P. Dale, Nov 04 2020 *)

Formula

n*a(n) = 6*a(n-1) + 4*(n-2)*a(n-2) for n > 1.
a(n) = A305031(n)/2 for n > 0.
Showing 1-2 of 2 results.

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

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