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 1-5 of 5 results.

A303346 Expansion of Product_{n>=1} ((1 + 2*x^n)/(1 - 2*x^n))^(1/2).

Original entry on oeis.org

1, 2, 4, 10, 18, 38, 72, 142, 260, 510, 940, 1814, 3362, 6490, 12112, 23466, 44114, 85766, 162516, 317190, 604806, 1184682, 2271248, 4461514, 8591784, 16916490, 32696708, 64496130, 125037142, 247007142, 480077432, 949510526, 1849375796, 3661330398, 7144215452
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2018

Keywords

Links

Crossrefs

Cf. A032302, A070933, A261584, A303307, A303345, A370709, A370713.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[Sqrt[-QPochhammer[-2, x] / (3*QPochhammer[2, x])], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+2*x^k)/(1-2*x^k))^(1/2)))

Formula

a(n) ~ 2^n / sqrt(c*Pi*n), where c = A048651 * A083864 = 1/2 * Product_{j>=1} (2^j-1)/(2^j+1) = 0.06056210400129025123042464659093375290492912341... - Vaclav Kotesovec, Apr 22 2018

A303387 Expansion of Product_{k>=1} ((1 - 4*x^k)/(1 + 4*x^k))^(1/4).

Original entry on oeis.org

1, -2, 0, -10, 22, -102, 244, -1270, 3360, -16886, 46160, -230670, 656550, -3238250, 9474684, -46289530, 138590342, -671116710, 2047182480, -9837322110, 30482926482, -145474988978, 456854466860, -2166890174370, 6884188144964, -32471461699594
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), this sequence (b=2), A303396 (b=3).
Cf. A303360, A303402.

Programs

  • Maple
    seq(coeff(series(mul(((1-4*x^k)/(1+4*x^k))^(1/4), k = 1..n), x, n+1), x, n), n=0..25); # Muniru A Asiru, Apr 23 2018
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1-4*x^k)/(1+4*x^k))^(1/4)))

Formula

a(n) ~ c * (-4)^n / n^(3/4), where c = (QPochhammer[-1, -1/4] / QPochhammer[-1/4])^(1/4) / Gamma(1/4) = 0.29599817925108933574246285.... - Vaclav Kotesovec, Apr 25 2018

A303439 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, 0, -2, 6, -6, 4, -6, 48, -118, 96, -78, 470, -810, 396, -3050, 11062, -12678, 7072, -21454, 80034, -201490, 218940, -200658, 1536724, -3268842, 2079312, -7013266, 23140282, -28227510, 24133668, -56293910, 288065712, -704485126, 629862288, -1176210654
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2018

Keywords

Links

Crossrefs

Cf. A303345, A303387, A303396, A303438.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[Sum[(1 - (-1)^j) / (j*(1 - 1/(2^(j-1)*x^j))), {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)
  • PARI
    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)))

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

A303397 Expansion of Product_{k>=1} (1 - 2*x^k)/(1 + 2*x^k).

Original entry on oeis.org

1, -4, 4, -4, 20, -36, 52, -116, 244, -500, 964, -1876, 3876, -7780, 15332, -30628, 61684, -123460, 246036, -491988, 985492, -1971284, 3939556, -7878068, 15762692, -31527428, 63041220, -126078916, 252185044, -504375460, 1008698036, -2017385268, 4034873268
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2018

Keywords

Crossrefs

Expansion of Product_{k>=1} (1 - b*x^k)/(1 + b*x^k): A002448 (b=1), this sequence (b=2), A303398 (b=3).
Cf. A261584, A303345.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 - 2*x^k)/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-2*x^k)/(1+2*x^k)))

Formula

a(n) ~ c * (-2)^n, where c = QPochhammer[-1, -1/2]/QPochhammer[-1/2] = 0.93943604828296530723602398257349307281... - Vaclav Kotesovec, Apr 25 2018
Showing 1-5 of 5 results.

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