A383315 Expansion of e.g.f. exp(-x/2) / (1-6*x)^(1/12).

1, 0, 3, 36, 675, 16632, 509085, 18626436, 793001097, 38511087120, 2101009734099, 127215916659540, 8465583820754907, 614101808094096744, 48230098800348987405, 4077120575169267005268, 369111206211249734907345, 35630377583888099367357984, 3653123185073359871950788963

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

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

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

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A383313, A383314.

Sequence in context: A296098 A224256 A223912 * A224184 A241996 A366004

Adjacent sequences: A383312 A383313 A383314 * A383316 A383317 A383318

Keyword

nonn

Author

Seiichi Manyama, Apr 23 2025

Status

approved