A386365 Expansion of e.g.f. (Sum_{k>=0} binomial(3*k,k) * x^k)^(1/3).
1, 1, 8, 118, 2536, 71600, 2504320, 104482000, 5063797760, 279579704320, 17322126976000, 1190107376057600, 89795437443712000, 7381088691027251200, 656522501105485414400, 62825541269686074880000, 6436003096247592964096000, 702751431134395346063360000
Offset: 0
Keywords
Programs
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Mathematica
nmax = 20; CoefficientList[Series[Sum[Binomial[3*k,k] * x^k, {k, 0, nmax}]^(1/3), {x, 0, nmax}], x] * Range[0,nmax]! (* Vaclav Kotesovec, Jul 19 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(3*k, k)*x^k)^(1/3)))
Formula
a(n) ~ Pi * 3^(3*n - 1/3) * n^(n - 1/3) / (Gamma(1/3)^2 * exp(n) * 2^(2*n - 1/2)). - Vaclav Kotesovec, Jul 19 2025
E.g.f.: B(x)^(1/3) where B(x) is the g.f. of A005809. - Georg Fischer, Jul 21 2025