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A052685
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Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4).
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1
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1, 1, 4, 24, 168, 1680, 18720, 252000, 3830400, 65681280, 1251936000, 26225337600, 599710003200, 14851444608000, 396138155212800, 11320537003776000, 345079573622784000, 11176410365632512000
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
D-finite recurrence: a(0)=1, a(1)=1, a(2)=4, a(3)=24, a(n) = n*a(n-1) + 2*n*(n-1)*a(n-2) - n*(n-1)*(n-2)*(n-3)*a(n-4).
a(n) = (n!/283)*Sum_{alpha=RootOf(1-z-2*Z^2+Z^4)} (27 + 112*alpha + 9*alpha^2 - 48*alpha^3)*alpha^(-1-n).
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MAPLE
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spec := [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Prod(Z, Z)))))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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b[n_]:= b[n]= If[n<4, n!-2*Boole[n==3], b[n-1] +2*b[n-2] -b[n-4]]; (* b=A052535 *)
With[{nn=20}, CoefficientList[Series[(1-x^2)/(1-x-2 x^2+x^4), {x, 0, nn}], x] Range[0, nn]!]
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PROG
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(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x^2)/(1-x-2*x^2+x^4) ))); // G. C. Greubel, Jun 03 2022
(SageMath)
@CachedFunction
if (n<4): return factorial(n) - 2*bool(n==3)
else: return b(n-1) + 2*b(n-2) - b(n-4)
def A052685(n): return factorial(n)*b(n)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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STATUS
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approved
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