A303342 -id:A303342 - cp's OEIS frontend

A303074 Expansion of Product_{n>=1} (1 + (9*x)^n)^(1/3).

1, 3, 18, 369, 1674, 31428, 266733, 3012714, 19924299, 319970007, 2688208641, 27248985549, 248061612240, 2597556114648, 25367004717831, 289880288735373, 2289952155529719, 23895509092285545, 252143223166599723, 2308267172943599733, 22389894059315522040

Offset: 0

Vaclav Kotesovec, Apr 18 2018

nonn

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A271236, A298994, A303342, A370739.

CoefficientList[Series[(QPochhammer[-1, 9*x]/2)^(1/3), {x, 0, 20}], x]

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

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

Original entry on oeis.org

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

A271236 G.f.: Product_{k>=1} 1/(1 - (9*x)^k)^(1/3).

Original entry on oeis.org

1, 3, 45, 450, 5805, 52326, 705591, 6190425, 77219325, 751178610, 8522919063, 80502824835, 975122402985, 8949951461925, 100088881882830, 1003346683458480, 10828622925516312, 104307212166072165, 1152197107898173875, 11048535008792967825, 119509353627934830327

Offset: 0

Vaclav Kotesovec, Apr 02 2016

nonn

In general, for h>=1, if g.f. = Product_{k>=1} 1/(1-(h^2*x)^k)^(1/h), then a(n) ~ h^(2*n) * exp(Pi*sqrt(2*n/(3*h))) / (2^(5*h+3) * 3^(h+1) * h^(h+1) * n^(3*h+1))^(1/(4*h)).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = 9^n. - Seiichi Manyama, Apr 20 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A298994, A303074, A303130, A303152, A303342, A370735.

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), this sequence (b=3), A303135 (b=4), A303136 (b=5).

nmax = 30; CoefficientList[Series[Product[1/(1 - (9*x)^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x]

N=99; x='x+O('x^N); Vec(prod(k=1, N, 1/(1-(9*x)^k)^(1/3))) \\ Altug Alkan, Apr 20 2018

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

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)).

cp's OEIS Frontend

A303074 Expansion of Product_{n>=1} (1 + (9*x)^n)^(1/3).

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

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A271236 G.f.: Product_{k>=1} 1/(1 - (9*x)^k)^(1/3).

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

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