A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

1, 2, 2, 20, 6, 108, 148, 776, -186, 5964, -4, 51032, -89700, 512120, -1259416, 6406032, -19733434, 78363148, -268823572, 1047941688, -3800035916, 14327505832, -52766730600, 199492430192, -746479735524, 2811936761016, -10588174502568, 40092283176560, -151796846803592

Offset: 0

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

• sign

Crossrefs

Cf. A032302, A370713.

Cf. A075900, A298994, A300581, A304961, A327550.

Programs

Mathematica

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

Formula

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

a(n) ~ (-1)^(n+1) * c * 4^n / n^(3/2), where c = QPochhammer(-1/2)^(1/2) / (2*sqrt(Pi)) = 0.31039710860287467176143051675437...