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A052591 E.g.f. x/((1-x)(1-x^2)). 4
0, 1, 2, 12, 48, 360, 2160, 20160, 161280, 1814400, 18144000, 239500800, 2874009600, 43589145600, 610248038400, 10461394944000, 167382319104000, 3201186852864000, 57621363351552000, 1216451004088320000, 24329020081766400000, 562000363888803840000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Stirling transform of 2*a(n) = [2,4,24,96,...] is A052841(n+1) = [2,6,38,270,...]. - Michael Somos, Mar 04 2004

From Emeric Deutsch, Jul 18 2009: (Start)

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.

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

(End)

LINKS

Table of n, a(n) for n=0..21.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 536

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)).

a(n) = (n+1)!/2 if n is odd; a(n) = n!n/2 if n is even. - Emeric Deutsch, Jul 18 2009

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

MAPLE

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

a:=n->(-1)*sum((-1)^k * (n-k+1) * n!, k=1..n) : seq(a(n), n=0..19); # Zerinvary Lajos, Jun 18 2007

a:=n->(n+1)!-sum((-1)^k*n!, k=0..n): seq(a(n)/2, n=0..19); # Zerinvary Lajos, Mar 25 2008

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

PROG

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

CROSSREFS

Cf. A052558. - Emeric Deutsch, Jul 18 2009

Sequence in context: A052569 A221663 A232663 * A029766 A088311 A338522

Adjacent sequences: A052588 A052589 A052590 * A052592 A052593 A052594

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified December 3 09:50 EST 2022. Contains 358517 sequences. (Running on oeis4.)