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A052564 Expansion of e.g.f. x*(1-x)/(1-2*x). 2

%I #27 Sep 08 2022 08:44:59

%S 0,1,2,12,96,960,11520,161280,2580480,46448640,928972800,20437401600,

%T 490497638400,12752938598400,357082280755200,10712468422656000,

%U 342798989524992000,11655165643849728000,419585963178590208000

%N Expansion of e.g.f. x*(1-x)/(1-2*x).

%C Partition the set {1,2,...,n} into an odd number of subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. - _Geoffrey Critzer_, Mar 05 2010

%H Vincenzo Librandi, <a href="/A052564/b052564.txt">Table of n, a(n) for n = 0..200</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=506">Encyclopedia of Combinatorial Structures 506</a>

%F E.g.f.: x*(1-x)/(1-2*x).

%F a(n) = 2*n*a(n-1), with a(0)=0, a(1)=1, a(2)=2.

%F a(n) = 2^(n-2) * n! for n>1.

%F a(n) = A002866(n) - A014297(n-2) for n>1. - _Geoffrey Critzer_, Mar 05 2010

%p spec := [S,{S=Prod(Z,Sequence(Prod(Z,Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t a = x/(1 - x); CoefficientList[Series[a/(1 - a^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* _Geoffrey Critzer_, Mar 05 2010 *)

%t Part[#,Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0]*#)&@CoefficientList[Series[x*(1-x)/(1-2x), {x, 0, 20}], x]// ExpandAll (* _Vincenzo Librandi_, Jan 04 2013 - after _Olivier Gérard_ in A213068 *)

%o (PARI) {a(n) = if(n<=1, n, 2^(n-2)*n!)}; \\ _G. C. Greubel_, May 05 2019

%o (Magma) [n le 1 select n else 2^(n-2)*Factorial(n): n in [0..20]]; // _G. C. Greubel_, May 05 2019

%o (Sage) [0,1]+[2^(n-2)*factorial(n) for n in (2..20)] # _G. C. Greubel_, May 05 2019

%o (PARI) my(x='x+O('x^20)); concat([0], Vec(serlaplace(x*(1-x)/(1-2*x)))) \\ _Felix Fröhlich_, May 05 2019

%Y Essentially the same as A014297.

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)