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.

A303381 Expansion of Product_{n>=1} ((1 + (8*x)^n)/(1 - (8*x)^n))^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 1526, 15228, 146676, 1217880, 10322982, 106429420, 886934236, 7632390312, 72137002428, 600860144728, 5351962341672, 51402944345520, 411439139563526, 3624067316629836, 33666668386023244, 279519776297893512, 2480351338204454484
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2018

Keywords

Comments

In general, if h>=1 and g.f. = Product_{k>=1} ((1 + (h*x)^k)/(1 - (h*x)^k))^(1/h), then a(n) ~ h^n * exp(Pi*sqrt(n/h)) /(2^(3/2 + 3/(2*h)) * h^(1/4 + 1/(4*h)) * n^(3/4 + 1/(4*h))). - Vaclav Kotesovec, Apr 23 2018

Links

Crossrefs

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A015128 (b=0), A303307 (b=1), A303361 (b=2), this sequence (b=3).
Cf. A303382.

Programs

  • Maple
    seq(coeff(series(mul(((1+(8*x)^k)/(1-(8*x)^k))^(1/8), k = 1..n), x, n+1), x, n), n = 0..25); # Muniru A Asiru, Apr 23 2018
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/8), {k, 1, nmax}], {x, 0, nmax}], x] * 8^Range[0, nmax] (* Vaclav Kotesovec, Apr 23 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+(8*x)^k)/(1-(8*x)^k))^(1/8)))

Formula

a(n) ~ 2^(3*n - 81/32) * exp(sqrt(n)*Pi/2^(3/2)) / n^(25/32). - Vaclav Kotesovec, Apr 23 2018

A303396 Expansion of Product_{k>=1} ((1 - 8*x^k)/(1 + 8*x^k))^(1/8).

Original entry on oeis.org

1, -2, 0, -42, 86, -1638, 4116, -76662, 218592, -3879766, 11965072, -205722702, 672706566, -11257625386, 38520382716, -630071416794, 2236375718918, -35864826630822, 131232962248816, -2068477295105214, 7767014381299026, -120556991420552658
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2018

Keywords

Links

Crossrefs

Expansion of Product_{k>=1} ((1 - 2^b*x^k)/(1 + 2^b*x^k))^(1/(2^b)): A002448 (b=0), A303345 (b=1), A303387 (b=2), this sequence (b=3).
Cf. A303382.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1-8*x^k)/(1+8*x^k))^(1/8)))

Formula

a(n) ~ c * (-8)^n / n^(7/8), where c = (QPochhammer[-1, -1/8] / QPochhammer[-1/8])^(1/8) / Gamma(1/8) = 0.14075750048358669653215841485... - Vaclav Kotesovec, Apr 25 2018

A303438 Expansion of Product_{k>=1} ((1 + 2^k*x^k)/(1 - 2^k*x^k))^(1/2^k).

Original entry on oeis.org

1, 2, 4, 10, 18, 38, 80, 158, 292, 630, 1260, 2470, 4922, 9706, 19392, 41010, 78466, 155494, 318764, 625670, 1238854, 2567666, 5106208, 10122522, 20022960, 40082154, 80027140, 163330106, 324201942, 643489014, 1306843568, 2592220110, 5081546084
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2018

Keywords

Comments

a(n) / 2^n tends to 1.2036... - Vaclav Kotesovec, Apr 25 2018

Links

Crossrefs

Cf. A015128, A303346, A303360, A303382, A303439.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + 2^k*x^k)/(1 - 2^k*x^k))^(1/2^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 24 2018 *)
nmax = 30; CoefficientList[Series[Exp[Sum[((-1)^j - 1) / (j*(1 - 1/(2^(j - 1)*x^j))), {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)
  • PARI
    my(N=66, x='x+O('x^N)); Vec(prod(k=1, N, ((1+2^k*x^k)/(1-2^k*x^k))^(1/2^k)))

Formula

G.f.: exp( Sum_{j>=1} ((-1)^j - 1) / (j*(1 - 1/(2^(j-1)*x^j))) ). - Vaclav Kotesovec, Apr 25 2018
Showing 1-3 of 3 results.

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