A303349 -id:A303349 - cp's OEIS frontend

A303348 Expansion of Product_{n>=1} (1 - 9*x^n)^(1/3).

1, -3, -12, -39, -246, -1578, -11487, -84054, -635781, -4893357, -38292969, -303553209, -2432865630, -19678331838, -160427322399, -1316796234933, -10872602692581, -90242886252945, -752488383572787, -6300541703215803, -52949782408528290

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

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

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

Expansion of Product_{n>=1} (1 - b^2*x^n)^(1/b): A010815 (b=1), A303347 (b=2), this sequence (b=3).

Cf. A303349, A303351.

seq(coeff(series(mul((1-9*x^k)^(1/3), k=1..n), x, n+1), x, n), n=0..25); # Muniru A Asiru, Apr 22 2018

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-9*x^k)^(1/3)))

a(n) ~ -c * 3^(2*n-1) / (Gamma(2/3) * n^(4/3)), where c = QPochhammer[1/9]^(1/3) = 0.95703379660353017269195329... - Vaclav Kotesovec, Apr 25 2018

A303353 Expansion of Product_{n>=1} 1/(1 + 9*x^n)^(1/3).

Original entry on oeis.org

1, -3, 15, -120, 915, -7086, 56661, -462405, 3819165, -31843110, 267610443, -2263491255, 19246265025, -164379723735, 1409306287470, -12122528944620, 104575462390842, -904411297029585, 7839310835762475, -68086561401745275, 592417977205534017

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

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

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

Expansion of Product_{n>=1} 1/(1 + b^2*x^n)^(1/b): A081362 (b=1), A303352 (b=2), this sequence (b=3).

Cf. A303349.

seq(coeff(series(mul(1/(1+9*x^k)^(1/3), k = 1..n), x, n+1), x, n), n = 0..25); # Muniru A Asiru, Apr 22 2018

nmax = 20; CoefficientList[Series[Product[1/(1 + 9*x^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)

a(n) ~ c * (-9)^n / n^(2/3), where c = 1 / (Gamma(1/3) * QPochhammer[-1/9]^(1/3)) = 0.361746646328749408912877789757526727... - Vaclav Kotesovec, Apr 25 2018

cp's OEIS Frontend

A303348 Expansion of Product_{n>=1} (1 - 9*x^n)^(1/3).

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