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 1-6 of 6 results.

A303538 Expansion of ((1 + 8*x)/(1 - 8*x))^(1/8).

Original entry on oeis.org

1, 2, 2, 44, 86, 1724, 4244, 80024, 223718, 4033132, 12260988, 213418728, 689489148, 11663520216, 39489621864, 652201870896, 2292944058246, 37099981422156, 134565259916012, 2138626858270408, 7964821656989332, 124595233474799752, 474734644904361112
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2018

Keywords

Links

Crossrefs

Cf. A063886, A303537.

Programs

  • Maxima
    a[n]:=if n<2 then n+1 else (2*a[n-1]+64*(n-2)*a[n-2])/n;
makelist(a[n],n,0,1000); /* Tani Akinari, Apr 29 2018 */
  • PARI
    N=66; x='x+O('x^N); Vec(((1+8*x)/(1-8*x))^(1/8))

Formula

a(n) ~ 2^(3*n + 1/8) / (Gamma(1/8) * n^(7/8)). - Vaclav Kotesovec, Apr 26 2018
n*a(n)-2*a(n-1)-64*(n-2)*a(n-2)=0. - Tani Akinari, Apr 29 2018

A304941 Expansion of ((1 + 4*x)/(1 - 4*x))^(3/4).

Original entry on oeis.org

1, 6, 18, 68, 246, 948, 3572, 13896, 53286, 208452, 807132, 3169080, 12346300, 48602760, 190150440, 750018448, 2943363078, 11627329764, 45736940364, 180897649368, 712881236052, 2822389182104, 11138924119512, 44137230865392, 174405194802524, 691557285091176
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), A304940 (m=2), this sequence (m=3), A081654 (m=4).

Programs

  • Magma
    [n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 8*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023
  • Mathematica
    CoefficientList[Series[((1+4x)/(1-4x))^(3/4),{x,0,30}],x] (* Harvey P. Dale, Oct 24 2020 *)
  • PARI
    N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(3/4))
  • SageMath
    @CachedFunction
def a(n): # a = A304941
    if n<2: return 6^n
    else: return 2*(3*a(n-1) + 8*(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) + 4^2*(n-2)*a(n-2) for n > 1.
a(n) ~ 2^(2*n + 3/4) / (Gamma(3/4) * n^(1/4)). - Vaclav Kotesovec, May 28 2018

A304933 a(0) = 0, a(1) = 1 and a(n) = 2*a(n-1)/(n-1) + 16*a(n-2) for n > 1.

Original entry on oeis.org

0, 1, 2, 18, 44, 310, 828, 5236, 14744, 87462, 255340, 1450460, 4349160, 23932220, 73268440, 393382440, 1224746032, 6447212294, 20354432076, 105417000268, 336767439560, 1720348748244, 5552121770888, 28030318314712, 91271367318096, 456091040311900
Offset: 0

Views

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).

Links

Crossrefs

a(n) = 2*a(n-1)/(n-1) + b^2*a(n-2): A001477 (b=1), A100071 (b=2), this sequence (b=4), A304934 (b=8).
Cf. A303537.

Formula

a(n) = n*A303537(n)/2.
G.f.: x/(1-4*x)^2 * ((1-4*x)/(1+4*x))^(3/4).

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).

A305608 Expansion of 1/2 * (((1 + 4*x)/(1 - 4*x))^(1/4) - 1).

Original entry on oeis.org

0, 1, 1, 6, 11, 62, 138, 748, 1843, 9718, 25534, 131860, 362430, 1840940, 5233460, 26225496, 76546627, 379247782, 1130801782, 5548263172, 16838371978, 81921368964, 252369171404, 1218709491944, 3802973638254, 18243641612476, 57570352319788
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 + k*x)/(1 - k*x))^(1/k) - 1): A001405(n-1) (k=2), this sequence (k=4), A305609 (k=8).
Cf. A303537.

Programs

  • Maple
    seq(coeff(series((1/2)*(((1+4*x)/(1-4*x))^(1/4)-1), x,35),x,n),n=0..30); # Muniru A Asiru, Jun 06 2018

Formula

n*a(n) = 2*a(n-1) + 16*(n-2)*a(n-2) for n > 1.
a(n) = A303537(n)/2 for n > 0.

A304915 Expansion of ((1 + 16*x)/(1 - 16*x))^(1/16).

Original entry on oeis.org

1, 2, 2, 172, 342, 26556, 67220, 4875160, 14125030, 973837420, 3087573628, 204536051176, 692771715836, 44412235657176, 158358513025896, 9874709152875568, 36706645561910150, 2234840966950941260, 8601116786415880940, 512801585354912006600, 2032977466125710169236
Offset: 0

Views

Author

Seiichi Manyama, May 21 2018

Keywords

Comments

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

Links

Crossrefs

Cf. A063886, A303537, A303538.

Programs

  • Mathematica
    CoefficientList[Series[((1+16x)/(1-16x))^(1/16),{x,0,30}],x] (* Harvey P. Dale, Jul 21 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(((1+16*x)/(1-16*x))^(1/16))

Formula

n*a(n) = 2*a(n-1) + 16^2*(n-2)*a(n-2) for n > 1.
a(n) ~ 2^(4*n + 1/16) / (Gamma(1/16) * n^(15/16)) * (1 - (-1)^n * sqrt(2 - sqrt(2 + sqrt(2))) * 2^(7/8) * Gamma(1/16)^2 / (64*Pi*n^(1/8))). - Vaclav Kotesovec, May 21 2018
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.