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A052693
Expansion of e.g.f. (1-x)/(1-3*x+x^3).
1
1, 2, 12, 102, 1176, 16920, 292320, 5891760, 135717120, 3517032960, 101268921600, 3207514464000, 110828037196800, 4148515981209600, 167232459621427200, 7222900141416960000, 332760193091149824000
OFFSET
0,2
LINKS
FORMULA
E.g.f.: (1-x)/(1-3*x+x^3).
Recurrence: a(0)=1, a(1)=2, a(2)=12 a(n+3) = 3*(n+3)*a(n+2) - (n+1)*(n+2)*(n+3)*a(n).
a(n) = (n!/9)*Sum_{alpha=RootOf(1 -3*Z +Z^3)} (2 - alpha + alpha^2)*alpha^(-1-n).
a(n) = n! * A052536(n). - G. C. Greubel, Jun 01 2022
MAPLE
spec := [S, {S=Sequence(Union(Z, Prod(Z, Union(Z, Sequence(Z)))))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
With[{nn=20}, CoefficientList[Series[(1-x)/(1-3x+x^3), {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Dec 17 2012 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x)/(1-3*x+x^3) ))); // G. C. Greubel, Jun 01 2022
(SageMath)
@CachedFunction
def A052536(n):
if (n<3): return factorial(n+1)
else: return 3*A052536(n-1) - A052536(n-3)
def A052693(n): return factorial(n)*A052536(n)
[A052693(n) for n in (0..40)] # G. C. Greubel, Jun 01 2022
CROSSREFS
Sequence in context: A123897 A351762 A302357 * A050621 A326237 A152254
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved