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EExpansion of e.g.f. (1-x)/(1-3x3*x+x^3).
G. C. Greubel, <a href="/A052693/b052693.txt">Table of n, a(n) for n = 0..375</a>
E.g.f.: -(-1+-x)/(1-3*x+x^3).
Recurrence: {a(0)=1, a(1)=2, a(2)=12 a(n^+3) = 3+6*n^2+11*(n+63)*a(n)+(2) -3* (n-9+1)*a(n+2)+a*(n+3)=0, *a(2n)=12}.
Sum(-1/9*(-2+_alpha^2-_alpha)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z+_Z^3))*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
(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
Cf. A052536.
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INRIA Algorithms Project, <a href="http://algoecs.inria.fr/ecsservices/ecsstructure?searchType=1&service=Search&searchTermsnbr=642">Encyclopedia of Combinatorial Structures 642</a>
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With[{nn=20}, CoefficientList[Series[(1-x)/(1-3x+x^3), {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Dec 17 2012 *)
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editing
INRIA Algorithms Project, <a href="http://algo.inria.fr/binecs/encyclopediaecs?searchType=1&=ECSnb&argsearchsearchTerms=642">Encyclopedia of Combinatorial Structures 642</a>