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A052591
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Expansion of e.g.f. x/((1-x)(1-x^2)).
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4
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0, 1, 2, 12, 48, 360, 2160, 20160, 161280, 1814400, 18144000, 239500800, 2874009600, 43589145600, 610248038400, 10461394944000, 167382319104000, 3201186852864000, 57621363351552000, 1216451004088320000, 24329020081766400000, 562000363888803840000
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OFFSET
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0,3
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COMMENTS
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Stirling transform of 2*a(n) = [2,4,24,96,...] is A052841(n+1) = [2,6,38,270,...]. - Michael Somos, Mar 04 2004
a(n) is the number of even fixed points in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 12'3, 132, 312, 213, 231, and 32'1, the even fixed points being marked. - Emeric Deutsch, Jul 18 2009
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LINKS
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FORMULA
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Recurrence: {a(1)=1, a(0)=0, (-n^3 - 5*n^2 - 8*n - 4)*a(n) + (-2-n)*a(n+1) + (n+1)*a(n+2) = 0}.
a(n) = ((1/4)*(-1)^(1-n) + (1/2)*n + 1/4)*n!.
E.g.f.: x/((1-x)*(1-x^2)).
a(n) = (n+1)!/2 if n is odd; a(n) = n!*n/2 if n is even.
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MAPLE
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spec := [S, {S=Prod(Z, Sequence(Z), Sequence(Prod(Z, Z)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
G(x):=x/(1-x)/(1-x^2): 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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PROG
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(PARI) a(n)=if(n<0, 0, n!*polcoeff(x/(1-x)/(1-x^2)+x*O(x^n), 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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