A052558 a(n) = n! *((-1)^n + 2*n + 3)/4.

1, 1, 4, 12, 72, 360, 2880, 20160, 201600, 1814400, 21772800, 239500800, 3353011200, 43589145600, 697426329600, 10461394944000, 188305108992000, 3201186852864000, 64023737057280000, 1216451004088320000, 26761922089943040000, 562000363888803840000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Stirling transform of (-1)^(n+1)*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 A258369. - Philippe Deléham, May 17 2005; corrected by Ilya Gutkovskiy, Jul 25 2018

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

From Gary W. Adamson, Apr 20 2009: (Start)

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

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

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. - Emeric Deutsch, Jul 18 2009

a(n) is also the number of permutations of [n+1] starting with an even number. - Olivier Gérard, Nov 07 2011

Links

• T. D. Noe, Table of n, a(n) for n=0..100
• A. Moreno Canadas, P. F. Fernandez Espinoza, and I. D. M. Gaviria, Categorification of some integer sequences via Kronecker Modules, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 500.

Crossrefs

Cf. A174548, A052841.

Cf. A002260. - Gary W. Adamson, Apr 20 2009

Cf. A052591. - Emeric Deutsch, Jul 18 2009

Cf. A052618, A077611, A199495. - Olivier Gérard, Nov 07 2011

Programs

GAP

List([0..30], n-> ((-1)^n +2*n +3)*Factorial(n)/4); # G. C. Greubel, May 07 2019

Magma

[((-1)^n +2*n +3)*Factorial(n)/4: n in [0..30]]; // G. C. Greubel, May 07 2019

Maple

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

Mathematica

Table[n!((-1)^n+2n+3)/4,{n,0,30}] (* Harvey P. Dale, Aug 16 2014 *)

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))

Sage

[((-1)^n +2*n +3)*factorial(n)/4 for n in (0..30)] # G. C. Greubel, May 07 2019

Formula

D-finite with recurrence a(n) = a(n-1) + (n^2-1)*a(n-2), with a(1)=1, a(0)=1.

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

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, Nov 14 2003

E.g.f.: 1/((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)/2 if n is even.

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

E.g.f.: G(0)/(1+x) where G(k) = 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)*(2*k+3)/(x*(2*k+3) + 2*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2012

Sum_{n>=0} 1/a(n) = e - 1/e = 2*sinh(1) (A174548). - Amiram Eldar, Jan 22 2023