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 1-3 of 3 results.

A305031 Expansion of ((1 + 2*x)/(1 - 2*x))^(3/2).

Original entry on oeis.org

1, 6, 18, 44, 102, 228, 500, 1080, 2310, 4900, 10332, 21672, 45276, 94248, 195624, 404976, 836550, 1724580, 3549260, 7293000, 14965236, 30669496, 62783448, 128388624, 262303132, 535422888, 1092063000, 2225728400, 4533175800, 9226818000, 18769219920, 38158909920
Offset: 0

Views

Author

Seiichi Manyama, May 24 2018

Keywords

Comments

Let ((1 + k*x)/(1 - k*x))^(m/k) = 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*x)/(1 - 2*x))^(m/2): A063886 (m=1), this sequence (m=3), A241204 (m=4).
Cf. A304941, A305032.

Programs

  • Magma
    [n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 2*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023
  • Mathematica
    CoefficientList[Series[((1+2*x)/(1-2*x))^(3/2), {x,0,40}], x] (* G. C. Greubel, Jun 07 2023 *)
  • PARI
    N=66; x='x+O('x^N); Vec(((1+2*x)/(1-2*x))^(3/2))
  • SageMath
    @CachedFunction
def a(n): # b = A305031
    if n<2: return 6^n
    else: return 2*(3*a(n-1) + 2*(n-2)*a(n-2))//n
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

n*a(n) = 6*a(n-1) + 4*(n-2)*a(n-2) for n > 1.
a(n) ~ 2^(n + 5/2) * sqrt(n/Pi). - Vaclav Kotesovec, May 28 2018

A304940 Expansion of ((1 + 4*x)/(1 - 4*x))^(1/2).

Original entry on oeis.org

1, 4, 8, 32, 96, 384, 1280, 5120, 17920, 71680, 258048, 1032192, 3784704, 15138816, 56229888, 224919552, 843448320, 3373793280, 12745441280, 50981765120, 193730707456, 774922829824, 2958796259328, 11835185037312, 45368209309696, 181472837238784
Offset: 0

Views

Author

Seiichi Manyama, May 22 2018

Keywords

Comments

Let ((1 + k*x)/(1 - k*x))^(m/k) = 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 + 4*x)/(1 - 4*x))^(m/4): A303537 (m=1), this sequence (m=2), A304941 (m=3), A081654 (m=4).
Cf. A063886.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/2))

Formula

n*a(n) = 4*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.
a(n) = 2^n * A063886(n).

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).

Links

Crossrefs

Cf. A304933, A304941.

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 1-3 of 3 results.

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