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A052688
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Expansion of e.g.f. x/((1-x)*(1-x^3)).
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1
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0, 1, 2, 6, 48, 240, 1440, 15120, 120960, 1088640, 14515200, 159667200, 1916006400, 31135104000, 435891456000, 6538371840000, 125536739328000, 2134124568576000, 38414242234368000, 851515702861824000
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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.: x/*(1-x)*(1-x^3)).
Recurrence: a(0)=0, a(1)=1, a(2)=2, (n+2)*a(n+3) = (n+3)*a(n+2) + (n+2)*(n+3)*a(n+1) + (n+1)*(n+2)*(n+3)^2*a(n).
a(n) = (n!/3)*(n + 1 - (1/3)*Sum_{alpha=RootOf(Z^2 + Z + 1)} (1 + 2*alpha)*_alpha^(-1-n)).
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MAPLE
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spec := [S, {S=Prod(Z, Sequence(Z), Sequence(Prod(Z, Z, Z)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
G(x):=x/(1-x)/(1-x^3): f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..19); # Zerinvary Lajos, Apr 03 2009
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[x/((1-x)(1-x^3)), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Dec 24 2018 *)
Table[n!*Floor[(n+2)/3], {n, 0, 40}] (* G. C. Greubel, Jun 02 2022 *)
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PROG
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(SageMath) [factorial(n)*((n+2)//3) for n in (0..40)] # G. C. Greubel, Jun 02 2022
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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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