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

0, 1, 2, 12, 96, 960, 11520, 161280, 2580480, 46448640, 928972800, 20437401600, 490497638400, 12752938598400, 357082280755200, 10712468422656000, 342798989524992000, 11655165643849728000, 419585963178590208000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 506

Crossrefs

Essentially the same as A014297.

Programs

Magma

[n le 1 select n else 2^(n-2)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 05 2019

Maple

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

Mathematica

a = x/(1 - x); CoefficientList[Series[a/(1 - a^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Geoffrey Critzer, Mar 05 2010 *)
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 *)
With[{nn=20},CoefficientList[Series[x (1-x)/(1-2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2025 *)

PARI

{a(n) = if(n<=1, n, 2^(n-2)*n!)}; \\ G. C. Greubel, May 05 2019

PARI

my(x='x+O('x^20)); concat([0], Vec(serlaplace(x*(1-x)/(1-2*x)))) \\ Felix Fröhlich, May 05 2019

Sage

[0,1]+[2^(n-2)*factorial(n) for n in (2..20)] # G. C. Greubel, May 05 2019

Formula

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

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

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

a(n) = A002866(n) - A014297(n-2) for n>1. - Geoffrey Critzer, Mar 05 2010