0,3

Stirling transform of -(-1)^n*a(n-1)=[1,-1,4,-12,72,-360,...] is A052841(n-1)=[1,0,2,6,38,270,...]. - Michael Somos Mar 04 2004

The Stirling transform of this sequence is A005649. - Philippe DELEHAM, May 17 2005

Ignoring reflections, this is the number of ways of connecting n+2 equally-spaced points on a circle with a path of n+1 line segments. See A030077 for the number of distinct lengths. - T. D. Noe, Jan 05 2007

Row sums of triangle A136581 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 09 2008

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 20 2009: (Start)

Signed: (+ - - + + - - + +,...) = eigensequence of triangle A002260

(1,2,3,...); "Crescendo" with alternate signs.

Example: -360 = (1, 1, -1, -4, 12, 71) dot (1, -2, 3, -4, 5, -6) = (1, -2, -3, 16, 60, -432). (End)

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009: (Start)

a(n) is the number of odd fixed points in all permutations of {1,2,...,n+1}, Example: a(2)=4 because we have 1'23', 1'32, 312, 213', 231, and 321, where the odd fixed points are marked.

a(n) = (n+1)! - A052591(n). (End)

T. D. Noe, Table of n, a(n) for n=0..100

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 500

Recurrence: {a(1)=1, a(0)=1, (-n^2-4*n-3)*a(n)+a(n+2)-a(n+1)}

(1/4*(-1)^(-n)+1/2*n+3/4)*n!

Let u(1)=1 u(n)=sum(k=1, n-1, u(k)*k*(-1)^(k-1)) then a(n)=abs(u(n+2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 14 2003

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

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009: (Start)

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

(End)

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

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

a:=n->sum(sum(n!*(-1)^j, j=0..20), k=0..n/2): seq(a(n), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 18 2007

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

(PARI) a(n)=if(n<0, 0, (1+n\2)*n!)

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

Cf. A136581.

A002260 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 20 2009]

A052591 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009]

Sequence in context: A009621 A013195 A166746 this_sequence A133666 A078628 A165261

Adjacent sequences: A052555 A052556 A052557 this_sequence A052559 A052560 A052561

easy,nonn

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