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A052558 a(n) = n! *((-1)^n + 2*n + 3)/4. 10
1, 1, 4, 12, 72, 360, 2880, 20160, 201600, 1814400, 21772800, 239500800, 3353011200, 43589145600, 697426329600, 10461394944000, 188305108992000, 3201186852864000, 64023737057280000, 1216451004088320000, 26761922089943040000, 562000363888803840000 (list; graph; refs; listen; history; text; internal format)
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

Row sums of triangle A136581. - Gary W. Adamson, Jan 09 2008

From Gary W. Adamson, 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)

From Emeric Deutsch, 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)

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 500

FORMULA

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)).

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

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) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 21 2012

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. A136581.

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: A013195 A323295 A166746 * A190340 A232325 A291487

Adjacent sequences:  A052555 A052556 A052557 * A052559 A052560 A052561

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified February 27 04:09 EST 2020. Contains 332299 sequences. (Running on oeis4.)