A303307 -id:A303307 - cp's OEIS frontend

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

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

Seiichi Manyama, Apr 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

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

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 *)

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

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

A303306 Expansion of Product_{n>=1} ((1 - (2x)^n)/(1 + (2x)^n))^(1/2).

Original entry on oeis.org

1, -2, -2, -4, 6, 4, 12, 56, 134, -108, 196, 328, -484, -88, -3752, -18576, 16838, -16460, -95340, -24408, -201036, -472584, 565544, 1424144, 1843356, -6632568, 10365224, 2317008, 49620088, 130484688, -4419664, 631241440, 761908550, -29690892, 329427380, -8889717144, 23673793860

Offset: 0

Seiichi Manyama, Apr 21 2018

sign

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

Cf. A054785, A303307, A303343.

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A303306(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 A303306(100)

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

A303361 Expansion of Product_{n>=1} ((1 + (4x)^n)/(1 - (4x)^n))^(1/4).

Original entry on oeis.org

1, 2, 10, 60, 262, 1372, 7044, 32760, 153670, 789676, 3659820, 17109320, 83073180, 381273240, 1786996424, 8604391920, 38832248902, 179714213580, 845485079580, 3834271942440, 17666638985652, 81920437065288, 370224975781560, 1685489994025360

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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

Cf. A303360.

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

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/4), {k, 1, nmax}], {x, 0, nmax}], x] * 4^Range[0, nmax] (* Vaclav Kotesovec, Apr 23 2018 *)

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

a(n) ~ 2^(2*n - 5/2) * exp(sqrt(n)*Pi/2) / n^(13/16). - Vaclav Kotesovec, Apr 23 2018

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

Seiichi Manyama, Apr 22 2018

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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.

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

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 *)

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

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

A303342 Expansion of Product_{k>=1} ((1 + (9x)^k) / (1 - (9x)^k))^(1/3).

Original entry on oeis.org

1, 6, 72, 1008, 10746, 130896, 1569456, 17371584, 192625128, 2260005462, 24725148912, 270748885392, 3027318848208, 32608207056528, 354309508944288, 3902606972751168, 41393526342215994, 443390745816982944, 4783687280410092984, 50532141192366275280

Offset: 0

Vaclav Kotesovec, Apr 22 2018

nonn

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

Cf. A271236, A303074, A303307.

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

a(n) ~ 3^(2*n) * exp(Pi*sqrt(n/3)) / (4 * 3^(1/3) * n^(5/6)).

A303440 Expansion of Product_{k>=1} ((1 + (2^kx)^k)/(1 - (2^kx)^k))^(1/2^k).

Original entry on oeis.org

1, 2, 10, 148, 8502, 2114924, 2151771524, 8800410198536, 144132802083312550, 9445021284412120235340, 2475898969479166225559648172, 2596153381122039693822323043973720, 10889040933791649565507987988056678914620

Offset: 0

Seiichi Manyama, Apr 24 2018

nonn

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

Cf. A303307, A303361, A303381, A303441.

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

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

Vaclav Kotesovec, Feb 29 2024

nonn

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)).

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.

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]

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)).

A303344 Expansion of Product_{n>=1} ((1 + (nx)^n)/(1 - (nx)^n))^(1/n).

Original entry on oeis.org

1, 2, 6, 28, 182, 1640, 19220, 278224, 4809942, 96598622, 2208156512, 56580566908, 1605518324884, 49963000166616, 1691615823420800, 61897541544248720, 2433873670903995990, 102341746590575878628, 4582360425862350559350, 217661837260679635780356

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Cf. A023881, A186633, A303307, A303343.

nmax = 20; CoefficientList[Series[Product[((1 + (k*x)^k)/(1 - (k*x)^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

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 22 2018

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

cp's OEIS Frontend

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

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A303306 Expansion of Product_{n>=1} ((1 - (2*x)^n)/(1 + (2*x)^n))^(1/2).

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A303361 Expansion of Product_{n>=1} ((1 + (4*x)^n)/(1 - (4*x)^n))^(1/4).

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

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A303342 Expansion of Product_{k>=1} ((1 + (9*x)^k) / (1 - (9*x)^k))^(1/3).

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A303440 Expansion of Product_{k>=1} ((1 + (2^k*x)^k)/(1 - (2^k*x)^k))^(1/2^k).

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

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A303344 Expansion of Product_{n>=1} ((1 + (n*x)^n)/(1 - (n*x)^n))^(1/n).

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

A303306 Expansion of Product_{n>=1} ((1 - (2x)^n)/(1 + (2x)^n))^(1/2).

A303361 Expansion of Product_{n>=1} ((1 + (4x)^n)/(1 - (4x)^n))^(1/4).

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

A303342 Expansion of Product_{k>=1} ((1 + (9x)^k) / (1 - (9x)^k))^(1/3).

A303440 Expansion of Product_{k>=1} ((1 + (2^kx)^k)/(1 - (2^kx)^k))^(1/2^k).

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

A303344 Expansion of Product_{n>=1} ((1 + (nx)^n)/(1 - (nx)^n))^(1/n).