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.

A304934 a(0) = 0, a(1) = 1 and a(n) = 2*a(n-1)/(n-1) + 64*a(n-2) for n > 1.

Original entry on oeis.org

0, 1, 2, 66, 172, 4310, 12732, 280084, 894872, 18149094, 61304940, 1173803004, 4136934888, 75812881404, 276427353048, 4891514031720, 18343552465968, 315349842088326, 1211087339244108, 20316955153568876, 79648216569893320, 1308249951485397396
Offset: 0

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Author

Seiichi Manyama, May 21 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 G.f. is x/(2*m) * d/dx ((1 + k*x)/(1 - k*x))^(m/k).

Crossrefs

a(n) = 2*a(n-1)/(n-1) + b^2*a(n-2): A001477 (b=1), A100071 (b=2), A304933 (b=4), this sequence (b=8).
Cf. A303538.

Formula

a(n) = n*A303538(n)/2.
G.f.: x/(1-8*x)^2 * ((1-8*x)/(1+8*x))^(7/8).

A304944 a(0) = 0, a(1) = 1 and a(n) = 6*a(n-1)/(n-1) + 16*a(n-2) for n > 1.

Original entry on oeis.org

0, 1, 6, 34, 164, 790, 3572, 16212, 71048, 312678, 1345220, 5809980, 24692600, 105305980, 443684360, 1875046120, 7848968208, 32944100998, 137210821092, 572842556332, 2376270786840, 9878362137364, 40842721771544, 169192718317336, 697620779210096
Offset: 0

Views

Author

Seiichi Manyama, May 22 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).

Crossrefs

Programs

  • Magma
    [n le 2 select n-1 else 2*(3*Self(n-1) + 8*(n-2)*Self(n-2))/(n-2): n in [1..40]]; // G. C. Greubel, Jun 07 2023
    
  • Mathematica
    CoefficientList[Series[x/((1-4*x)^(7/4)*(1+4*x)^(1/4)), {x,0,40}], x] (* G. C. Greubel, Jun 07 2023 *)
  • SageMath
    @CachedFunction
    def a(n): # b = A304944
        if n<2: return n
        else: return 2*(3*a(n-1) + 8*(n-1)*a(n-2))//(n-1)
    [a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

a(n) = n*A304941(n)/6.
G.f.: x/(1-4*x)^2 * ((1-4*x)/(1+4*x))^(1/4).
Showing 1-2 of 2 results.