A379825 a(n) = [x^n] x/(12*x^2 - 6*x + 1).

0, 1, 6, 24, 72, 144, 0, -1728, -10368, -41472, -124416, -248832, 0, 2985984, 17915904, 71663616, 214990848, 429981696, 0, -5159780352, -30958682112, -123834728448, -371504185344, -743008370688, 0, 8916100448256, 53496602689536, 213986410758144, 641959232274432

Offset: 0

Author

Peter Luschny, Jan 04 2025

Keywords

• sign
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-12).

Crossrefs

Cf. A057083, A000748, A190965.

Programs

Maple

w := sqrt(-3): a := n -> (w/6)*((3 - w)^n - (3 + w)^n):
seq(simplify(a(n)), n = 0..28);
# Alternative:
a := proc(n) option remember; if n < 2 then n else 6*(a(n - 1) - 2*a(n - 2)) fi end:
seq(a(n), n = 0..28);

Mathematica

LinearRecurrence[{6,-12},{0,1},29] (* James C. McMahon, Jan 05 2025 *)

Formula

a(n) = n! * [x^n] exp(3*x)*sin(sqrt(3)*x)/sqrt(3).

a(n) = (w/6)*((3 - w)^n - (3 + w)^n) where w = sqrt(-3).

a(n) = 6*a(n - 1) - 12*a(n - 2) for n >= 2.

a(n) = 2^n*3^((n - 1)/2)*sin((Pi*n)/6).

a(n) = 2^(n-1)*A057083(n-1) = 2^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, sqrt(3)/2) for n >= 1.