A052591 Expansion of e.g.f. x/((1-x)(1-x^2)).

0, 1, 2, 12, 48, 360, 2160, 20160, 161280, 1814400, 18144000, 239500800, 2874009600, 43589145600, 610248038400, 10461394944000, 167382319104000, 3201186852864000, 57621363351552000, 1216451004088320000, 24329020081766400000, 562000363888803840000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 536.

Crossrefs

Cf. A008619, A052841.

Cf. A052558. - Emeric Deutsch, Jul 18 2009

Cf. A001620, A073743, A099284.

Programs

Maple

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

PARI

a(n)=if(n<0,0,n!*polcoeff(x/(1-x)/(1-x^2)+x*O(x^n),n))

Formula

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)).

From Emeric Deutsch, Jul 18 2009: (Start)

a(n) = (n+1)!/2 if n is odd; a(n) = n!*n/2 if n is even.

a(n) = (n+1)! - A052558(n). (End)

a(n) = n!*A008619(n-1), n > 1. - R. J. Mathar, Nov 27 2011

Sum_{n>=1} 1/a(n) = 2*(CoshIntegral(1) + cosh(1) - gamma - 1) = 2*(A099284 + A073743 - A001620 - 1). - Amiram Eldar, Jan 22 2023