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 1-7 of 7 results.

A303345 Expansion of Product_{k>=1} ((1 - 2*x^k)/(1 + 2*x^k))^(1/2).

Original entry on oeis.org

1, -2, 0, -2, 6, -6, 12, -22, 48, -94, 160, -318, 622, -1210, 2268, -4482, 8678, -16998, 32632, -64366, 124674, -245866, 476108, -940866, 1829148, -3617066, 7040112, -13937530, 27186810, -53857062, 105196572, -208546726, 407944704, -809175966, 1584713040
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2018

Keywords

Links

Crossrefs

Cf. A303306, A303346, A303439.
Expansion of Product_{k>=1} ((1 - 2^b*x^k)/(1 + 2^b*x^k))^(1/(2^b)): A002448 (b=0), this sequence (b=1), A303387 (b=2), A303396 (b=3).

Programs

  • Maple
    seq(coeff(series(mul(((1-2*x^k)/(1+2*x^k))^(1/2), k = 1..n), x, n+1), x, n), n=0..40); # Muniru A Asiru, 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) ~ c * (-2)^n / sqrt(Pi*n), where c = (QPochhammer[-1, -1/2] / QPochhammer[-1/2])^(1/2) = 0.96924509195711964009315.... - Vaclav Kotesovec, Apr 25 2018

A303360 Expansion of Product_{n>=1} ((1 + 4*x^n)/(1 - 4*x^n))^(1/4).

Original entry on oeis.org

1, 2, 4, 18, 34, 166, 384, 1902, 4756, 24022, 64284, 321542, 899658, 4455690, 12888944, 63185250, 187513426, 910880550, 2759413788, 13295839638, 40967821494, 195979968882, 612569599440, 2911592648458, 9213101043936, 43538337410474, 139246245625364
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2018

Keywords

Links

Crossrefs

Expansion of Product_{n>=1} ((1 + 2^b*x^n)/(1 - 2^b*x^n))^(1/(2^b)): A015128 (b=0), A303346 (b=1), this sequence (b=2).
Cf. A303361, A303391, A067855, A303350, A303392.

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..35); # Muniru A Asiru, Apr 22 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + 4*x^k)/(1 - 4*x^k))^(1/4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, 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) = 0.3885547372628... - Vaclav Kotesovec, Apr 23 2018

A303382 Expansion of Product_{n>=1} ((1 + 8*x^n)/(1 - 8*x^n))^(1/8).

Original entry on oeis.org

1, 2, 4, 50, 98, 1830, 4576, 83950, 236500, 4211766, 12903260, 222377926, 723722602, 12136867530, 41382435824, 678060771778, 2400028798290, 38546050682278, 140724756748476, 2220907298526934, 8323586858891766, 129340015891714962, 495838256186203600
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2018

Keywords

Links

Crossrefs

Expansion of Product_{n>=1} ((1 + 2^b*x^n)/(1 - 2^b*x^n))^(1/(2^b)): A015128 (b=0), A303346 (b=1), A303360 (b=2), this sequence (b=3).
Cf. A303381.

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 = 25; CoefficientList[Series[Product[((1 + 8*x^k)/(1 - 8*x^k))^(1/8), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 23 2018 *)
nmax = 30; CoefficientList[Series[(-7*QPochhammer[-8, x] / (9*QPochhammer[8, x]))^(1/8), {x, 0, nmax}], x] (* 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) ~ c * 8^n / n^(7/8), where c = (QPochhammer[-1, 1/8] / QPochhammer[1/8])^(1/8) / Gamma(1/8) = 0.15003359366795844474467456149... - Vaclav Kotesovec, Apr 23 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

A370749 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^(1/4).

Original entry on oeis.org

1, 2, 6, 28, 70, 300, 892, 3544, 9990, 43340, 127988, 546120, 1651356, 7227896, 22414008, 99344944, 312879302, 1396285452, 4486205892, 20057934312, 65293087284, 292353604136, 963327294536, 4308913730256, 14340603113372, 64059675491512, 215075154021384, 958968160741328
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 29 2024

Keywords

Crossrefs

Cf. A261584, A303346, A370750.
Cf. A032302, A070933.
Cf. A370736, A370732.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 2*(2*x)^k)/(1 - 2*(2*x)^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + 2*(2*x)^k)/(1 - 2*(2*x)^k))^(1/4).
a(n) ~ QPochhammer(-1, 1/2)^(1/4) * 4^n / (Gamma(1/4) * QPochhammer(1/2)^(1/4) * n^(3/4)).

A370750 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^(1/3).

Original entry on oeis.org

1, 12, 180, 3852, 50436, 947052, 14087844, 245858652, 3531115620, 64019229660, 950199749748, 16959724619004, 256888616329044, 4642974930688812, 71716402072904724, 1308491345357401068, 20501966472318764388, 376230182366985289164, 5987314157007778195716, 110286515004790197907836
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 29 2024

Keywords

Crossrefs

Cf. A261584, A303346, A370749.
Cf. A032302, A070933.
Cf. A370716, A370715.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[(1 + 2*(9*x)^k)/(1 - 2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + 2*(9*x)^k)/(1 - 2*(9*x)^k))^(1/3).
a(n) ~ QPochhammer(-1, 1/2)^(1/3) * 18^n / (Gamma(1/3) * QPochhammer(1/2)^(1/3) * n^(2/3)).

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3*x^k)/(1 - 3*x^k))^(1/3).

Original entry on oeis.org

1, 6, 36, 360, 1998, 18792, 121176, 1123632, 7537860, 72078174, 510702408, 4896308088, 35923749480, 345406994280, 2600934294816, 24985346997888, 191735328374478, 1838307293836560, 14317601666954364, 136953233511162840, 1079293961918593800, 10299943344889922832
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 29 2024

Keywords

Comments

In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} ((1 + d*x^k)/(1 - d*x^k))^(1/m), then a(n) ~ QPochhammer(-1, 1/d)^(1/m) * d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

Crossrefs

Cf. A303390 (d=3,m=1), A370751 (d=3,m=2), A370752 (d=3,m=3).
Cf. A261584 (d=2,m=1), A303346 (d=2,m=2), A370750 (d=2,m=3), A370749 (d=2,m=4).
Cf. A015128 (d=1,m=1), A303307 (d=1,m=2), A303342 (d=1,m=3).
Cf. A032308, A242587.
Cf. A370712, A370710.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k)/(1 - 3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(3*x)^k)/(1 - 3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + 3*(3*x)^k)/(1 - 3*(3*x)^k))^(1/3).
a(n) ~ QPochhammer(-1, 1/3)^(1/3) * 9^n / (Gamma(1/3) * QPochhammer(1/3)^(1/3) * n^(2/3)).
Showing 1-7 of 7 results.

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