A383315 - OEIS

%I #14 Feb 11 2026 13:43:47

%S 1,0,3,36,675,16632,509085,18626436,793001097,38511087120,

%T 2101009734099,127215916659540,8465583820754907,614101808094096744,

%U 48230098800348987405,4077120575169267005268,369111206211249734907345,35630377583888099367357984,3653123185073359871950788963

%N Expansion of e.g.f. exp(-x/2) / (1-6*x)^(1/12).

%F a(n) = (-1)^n * n! * Sum_{k=0..n} 3^k * (1/2)^(n-2*k) * binomial(-1/12,k)/(n-k)!.

%F a(n) = 3*(n-1) * (2*a(n-1) + a(n-2)) for n > 1.

%F a(n) ~ Pi * (2 - sqrt(3))^(1/4) * 2^(n + 1/2) * 3^(n - 3/8) * n^(n - 5/12) / (Gamma(1/3) * Gamma(1/4) * exp(n + 1/12)). - _Vaclav Kotesovec_, Apr 23 2025

%t With[{nn=20},CoefficientList[Series[Exp[-x/2]/Surd[1-6x,12],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 11 2026 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/2)/(1-6*x)^(1/12)))

%Y Cf. A383313, A383314.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 23 2025