A303306 -id:A303306 - cp's OEIS frontend

A303307 Expansion of Product_{n>=1} ((1 + (2x)^n)/(1 - (2x)^n))^(1/2).

1, 2, 6, 20, 54, 156, 444, 1192, 3174, 8620, 22516, 58392, 151996, 387352, 984888, 2507088, 6270854, 15659724, 39067588, 96454072, 237663444, 584266696, 1425921992, 3470869296, 8431325916, 20380759544, 49122457608, 118178637040, 283150466232, 676768288176

Offset: 0

Seiichi Manyama, Apr 21 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3000

Cf. A054785, A303306, A303342.

CoefficientList[Series[Sqrt[QPochhammer[-1, 2*x] / (2*QPochhammer[2*x])], {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 21 2018 *)

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A303307(n)
  ary = [1]
  a = (0..n).map{|i| 2 ** (i - 1) * (s(2 * i) - s(i))}
  (1..n).each{|i| ary << (1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
  ary
end
p A303307(100)

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 2^(k-1) * A054785(k) * a(n-k) for n > 0.

a(n) ~ 2^(n - 21/8) * exp(Pi*sqrt(n/2)) / n^(7/8). - Vaclav Kotesovec, Apr 21 2018

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

Seiichi Manyama, Apr 22 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..3000

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).

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

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1-2*x^k)/(1+2*x^k))^(1/2)))

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

A303394 Expansion of Product_{n>=1} ((1 - (4x)^n)/(1 + (4x)^n))^(1/4).

Original entry on oeis.org

1, -2, -6, -28, -26, -156, -476, 968, 11526, -16172, 139724, 791928, 1315548, 12772840, 31004424, -105335920, 1058225606, 2239259700, -3700870212, 29301955992, -4944685836, -526686535112, 1134044530040, 4057865621232, -13063873857124, -113573062924024

Offset: 0

Seiichi Manyama, Apr 23 2018

sign

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A002448 (b=0), A303306 (b=1), this sequence (b=2), A303395 (b=3).

Cf. A303361.

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1-(4*x)^k)/(1+(4*x)^k))^(1/4)))

A303395 Expansion of Product_{n>=1} ((1 - (8x)^n)/(1 + (8x)^n))^(1/8).

Original entry on oeis.org

1, -2, -14, -140, -586, -5628, -44492, -187864, -482906, -17262828, -37958116, 65242328, -2453533124, 21817485480, 113877127592, -2570293967536, 28064994368262, 169252831757492, -188223954450804, 12624245609040632, 54083158292451540, -158891267358816264

Offset: 0

Seiichi Manyama, Apr 23 2018

sign

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A002448 (b=0), A303306 (b=1), A303394 (b=2), this sequence (b=3).

Cf. A303381.

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

A303343 Expansion of Product_{n>=1} ((1 - (nx)^n)/(1 + (nx)^n))^(1/n).

Original entry on oeis.org

1, -2, -2, -12, -90, -968, -12764, -200464, -3674378, -76958942, -1814783184, -47618072204, -1376556963244, -43481881639608, -1490306874257184, -55091497907730000, -2185104061817592618, -92567886910879671396, -4171638481112174023226

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

Cf. A303306, A303344.

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

a(n) ~ -2 * n^(n-1). - Vaclav Kotesovec, Apr 25 2018

G.f.: exp(Sum_{k>=1} (sigma_k(k) - sigma_k(2*k))*x^k/(2^(k-1)*k)). - Ilya Gutkovskiy, Apr 14 2019

A303441 Expansion of Product_{k>=1} ((1 - (2^kx)^k)/(1 + (2^kx)^k))^(1/2^k).

Original entry on oeis.org

1, -2, -6, -116, -7914, -2080044, -2143234268, -8791784539464, -144097582730402202, -9444444682320960008908, -2475861188239098437867827380, -2596143477450571483513272599120280, -10889030549158315918778789768740624543108

Offset: 0

Seiichi Manyama, Apr 24 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..58

Cf. A303306, A303394, A303395, A303440.

N=20; x='x+O('x^N); Vec(prod(k=1, N, ((1-(2^k*x)^k)/(1+(2^k*x)^k))^(1/2^k)))

a(n) ~ -2^(n^2 - n + 1). - Vaclav Kotesovec, Apr 25 2018

cp's OEIS Frontend

A303307 Expansion of Product_{n>=1} ((1 + (2*x)^n)/(1 - (2*x)^n))^(1/2).

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

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A303394 Expansion of Product_{n>=1} ((1 - (4*x)^n)/(1 + (4*x)^n))^(1/4).

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A303395 Expansion of Product_{n>=1} ((1 - (8*x)^n)/(1 + (8*x)^n))^(1/8).

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A303343 Expansion of Product_{n>=1} ((1 - (n*x)^n)/(1 + (n*x)^n))^(1/n).

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A303441 Expansion of Product_{k>=1} ((1 - (2^k*x)^k)/(1 + (2^k*x)^k))^(1/2^k).

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A303307 Expansion of Product_{n>=1} ((1 + (2x)^n)/(1 - (2x)^n))^(1/2).

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

A303394 Expansion of Product_{n>=1} ((1 - (4x)^n)/(1 + (4x)^n))^(1/4).

A303395 Expansion of Product_{n>=1} ((1 - (8x)^n)/(1 + (8x)^n))^(1/8).

A303343 Expansion of Product_{n>=1} ((1 - (nx)^n)/(1 + (nx)^n))^(1/n).

A303441 Expansion of Product_{k>=1} ((1 - (2^kx)^k)/(1 + (2^kx)^k))^(1/2^k).