A370739 -id:A370739 - cp's OEIS frontend

A298994 Expansion of Product_{n>=1} (1 + (4*x)^n)^(1/2).

1, 2, 6, 52, 134, 956, 4124, 20008, 73158, 439660, 1874612, 8350808, 37583004, 169862616, 779948152, 3774085968, 15435601222, 69542934604, 329825707332, 1403190752632, 6313190864052, 29079505547912, 126937389732872, 552273916408368, 2477249228318748

Offset: 0

Seiichi Manyama, Jan 31 2018

nonn

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

Cf. A271235, A298411, A298993, A303074, A370739.

CoefficientList[Series[Sqrt[QPochhammer[-1, 4*x]/2], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 18 2018 *)

Convolution inverse of A298993.
a(n) ~ 2^(2*n - 2) * exp(Pi*sqrt(n/6)) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 18 2018
Sum_{k=0..n} a(k)*a(n-k) = 4^n * A000009(n). - Vaclav Kotesovec, Jun 07 2025

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

Original entry on oeis.org

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)).

A370738 a(n) = 8^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/4).

Original entry on oeis.org

1, 6, -6, 1428, -13146, 280788, -3785820, 93142824, -1851272826, 37533646212, -765409050420, 16617464296728, -357906128318628, 7730398360992840, -168750405673899000, 3719099270015849040, -82288133754592611642, 1828585054153956768612, -40828782977534929747524, 915461326204911371035320

Offset: 0

Vaclav Kotesovec, Feb 28 2024

sign

Cf. A032308 (m=1), A370711 (m=2), A370712 (m=3), A370739 (m=5).

nmax = 20; CoefficientList[Series[Product[1+3*x^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 8^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1+3*(8*x)^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + 3*(8*x)^k)^(1/4).

a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/4) * 24^n / (4 * Gamma(3/4) * n^(5/4)).

cp's OEIS Frontend

A298994 Expansion of Product_{n>=1} (1 + (4*x)^n)^(1/2).

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

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A370738 a(n) = 8^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/4).

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