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

0,3

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.

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

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 *)

Prog

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

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

Sequence in context: A356667 A323295 A166746 * A369285 A190340 A232325

Adjacent sequences: A052555 A052556 A052557 * A052559 A052560 A052561

Keyword

easy,nonn

Author

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

Status

approved