cp's OEIS Frontend

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

%I A052558 #71 Dec 20 2024 16:13:09
%S A052558 1,1,4,12,72,360,2880,20160,201600,1814400,21772800,239500800,
%T A052558 3353011200,43589145600,697426329600,10461394944000,188305108992000,
%U A052558 3201186852864000,64023737057280000,1216451004088320000,26761922089943040000,562000363888803840000
%N A052558 a(n) = n! *((-1)^n + 2*n + 3)/4.
%C A052558 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
%C A052558 The Stirling transform of this sequence is A258369. - _Philippe Deléham_, May 17 2005; corrected by _Ilya Gutkovskiy_, Jul 25 2018
%C A052558 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
%C A052558 From _Gary W. Adamson_, Apr 20 2009: (Start)
%C A052558 Signed: (+ - - + + - - + +, ...) = eigensequence of triangle A002260.
%C A052558 Example: -360 = (1, 1, -1, -4, 12, 71) dot (1, -2, 3, -4, 5, -6) = (1, -2, -3, 16, 60, -432). (End)
%C A052558 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
%C A052558 a(n) is also the number of permutations of [n+1] starting with an even number. - _Olivier Gérard_, Nov 07 2011
%H A052558 T. D. Noe, <a href="/A052558/b052558.txt">Table of n, a(n) for n=0..100</a>
%H A052558 A. Moreno Canadas, P. F. Fernandez Espinoza, and I. D. M. Gaviria, <a href="http://dx.doi.org/10.17654/NT038040339&#34;">Categorification of some integer sequences via Kronecker Modules</a>, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
%H A052558 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=500">Encyclopedia of Combinatorial Structures 500</a>.
%F A052558 D-finite with recurrence a(n) = a(n-1) + (n^2-1)*a(n-2), with a(1)=1, a(0)=1.
%F A052558 a(n) = ((-1)^n + 2*n + 3)*n!/4.
%F A052558 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
%F A052558 E.g.f.: 1/((1-x)*(1-x^2)).
%F A052558 From _Emeric Deutsch_, Jul 18 2009: (Start)
%F A052558 a(n) = (n+1)!/2 if n is odd; a(n) = n!(n+2)/2 if n is even.
%F A052558 a(n) = (n+1)! - A052591(n). (End)
%F A052558 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
%F A052558 Sum_{n>=0} 1/a(n) = e - 1/e = 2*sinh(1) (A174548). - _Amiram Eldar_, Jan 22 2023
%p A052558 spec := [S,{S=Prod(Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052558 Table[n!((-1)^n+2n+3)/4,{n,0,30}] (* _Harvey P. Dale_, Aug 16 2014 *)
%o A052558 (PARI) a(n)=if(n<0,0,(1+n\2)*n!)
%o A052558 (PARI) a(n)=if(n<0, 0, n!*polcoeff(1/(1-x)/(1-x^2)+x*O(x^n), n))
%o A052558 (Magma) [((-1)^n +2*n +3)*Factorial(n)/4: n in [0..30]]; // _G. C. Greubel_, May 07 2019
%o A052558 (Sage) [((-1)^n +2*n +3)*factorial(n)/4 for n in (0..30)] # _G. C. Greubel_, May 07 2019
%o A052558 (GAP) List([0..30], n-> ((-1)^n +2*n +3)*Factorial(n)/4); # _G. C. Greubel_, May 07 2019
%Y A052558 Cf. A174548, A052841.
%Y A052558 Cf. A002260. - _Gary W. Adamson_, Apr 20 2009
%Y A052558 Cf. A052591. - _Emeric Deutsch_, Jul 18 2009
%Y A052558 Cf. A052618, A077611, A199495. - _Olivier Gérard_, Nov 07 2011
%K A052558 easy,nonn
%O A052558 0,3
%A A052558 encyclopedia(AT)pommard.inria.fr, Jan 25 2000