A370327 E.g.f.: exp(Sum_{k>=1} binomial(3*k,k) * x^k).

1, 3, 39, 801, 22329, 783963, 33142959, 1637118297, 92464016913, 5874794703027, 414582627839319, 32165264232976977, 2720659595107779081, 249128649448324321419, 24549819503580405230751, 2590147977759401893135497, 291286776584800990966021281, 34781064869030286562513208163

Offset: 0

Author

Vaclav Kotesovec, Feb 15 2024

Keywords

• nonn

Crossrefs

Cf. A005809, A229451, A370055, A370326.

Programs

Mathematica

CoefficientList[Series[Exp[Sum[Binomial[3*k, k]*x^k, {k, 1, 20}]], {x, 0, 20}], x] * Range[0, 20]!
CoefficientList[Series[Exp[2*Cos[ArcCos[1 - 27*x/2]/6] / Sqrt[4 - 27*x] - 1], {x, 0, 20}], x] * Range[0, 20]!

Formula

E.g.f.: exp(2*cos(arccos(1 - 27*x/2)/6) / sqrt(4 - 27*x) - 1).

a(n) ~ 3^(3*n - 1/3) * exp((3/2)^(4/3)*n^(1/3) - n - 5/6) * n^(n - 1/3) / 2^(2*n + 1/6).