A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

1, 3, 0, 99, -270, 2430, -10287, 105462, -750141, 5702481, -42623901, 347424633, -2779077762, 22353287634, -181730796723, 1493711042589, -12321529794261, 102125312638713, -850797139405887, 7120067746384863, -59800770201017934, 503922807927384129, -4259721779079782751

Offset: 0

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

• sign

Crossrefs

Cf. A032308, A370710.

Cf. A300579, A344062.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[(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), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(QPochhammer[-3, x]/4)^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

Formula

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

a(n) ~ (-1)^(n+1) * c * 9^n / n^(4/3), where c = QPochhammer(-1/3)^(1/3) / (3*Gamma(2/3)) = 0.26286302373105271371291957730496322329245126572...