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 1-4 of 4 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

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

A303402 Expansion of Product_{k>=1} (1 - 4*x^k)/(1 + 4*x^k).

Original entry on oeis.org

1, -8, 24, -72, 344, -1416, 5400, -21576, 87000, -348296, 1390872, -5560776, 22253784, -89025672, 356055960, -1424186568, 5696931032, -22787865096, 91150729368, -364602357960, 1458412314456, -5833651510536, 23334594559128, -93338369011272, 373353522099288
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), A303397 (b=2), A303398 (b=3), this sequence (b=4).
Cf. A303387, A303391.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - 4*x^k)/(1 + 4*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, (1-4*x^k)/(1+4*x^k)))

Formula

a(n) ~ c * (-4)^n, where c = QPochhammer[-1, -1/4]/QPochhammer[-1/4] = 1.3264181585010678966173808329272239860188791629... - Vaclav Kotesovec, Apr 25 2018
Showing 1-4 of 4 results.

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