A370711 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/2).

1, 6, 6, 348, -570, 12084, -31332, 780792, -6111930, 65506884, -599418444, 6707736456, -69508986852, 738378468744, -7878832564872, 85524000547056, -929068361832378, 10158667075255524, -111690827626777788, 1234592278534799592, -13700571880245603276, 152613494540593338264

Offset: 0

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

• sign

Crossrefs

Cf. A032308, A370714.

Cf. A300579, A344062.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(4*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-3, x]/4], {x, 0, nmax}], x] * 4^Range[0, nmax]

Formula

G.f.: Product_{k>=1} (1 + 3*(4*x)^k)^(1/2).

a(n) ~ (-1)^(n+1) * c * 12^n / n^(3/2), where c = QPochhammer(-1/3)^(1/2) / (2*sqrt(Pi)) = 0.311283382185276347775502154581850436407169685238...