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%I #78 Apr 18 2023 09:44:12
%S 0,2,4,12,48,240,1440,10080,80640,725760,7257600,79833600,958003200,
%T 12454041600,174356582400,2615348736000,41845579776000,
%U 711374856192000,12804747411456000,243290200817664000,4865804016353280000,102181884343418880000,2248001455555215360000
%N a(0) = 0; a(n+1) = 2*n! (n >= 0).
%C For n >= 1 a(n) is the size of the centralizer of a transposition in the symmetric group S_(n+1). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
%C For n > 0, a(n) = n! - A062119(n-1) = number of permutations of length n that have two specified elements adjacent. For example, a(4) = 12 as of the 24 permutations, 12 have say 1 and 2 adjacent: 1234, 2134, 1243, 2143, 3124, 3214, 4123, 4213, 3412, 3421, 4312, 4321. - _Jon Perry_, Jun 08 2003
%C With different offset, denominators of certain sums computed by Ramanujan.
%C From _Michael Somos_, Mar 04 2004: (Start)
%C Stirling transform of a(n) = [2, 4, 12, 48, 240, ...] is A000629(n) = [2, 6, 26, 150, 1082, ...].
%C Stirling transform of a(n-1) = [1, 2, 4, 12, 48, ...] is A007047(n-1) = [1, 3, 11, 51, 299, ...].
%C Stirling transform of a(n) = [1, 4, 12, 48, 240, ...] is A002050(n) = [1, 5, 25, 149, 1081, ...].
%C Stirling transform of 2*A006252(n) = [2, 2, 4, 8, 28, ...] is a(n) = [2, 4, 12, 48, 240, ...].
%C Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 2*A005649(n) = [4, 16, 88, 616, ...].
%C Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 4*A083410(n) = [4, 16, 88, 616, ...]. (End)
%C Number of {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
%C Permanent of the (0, 1)-matrices with (i, j)-th entry equal to 0 if and only if it is in the border but not the corners. The border of a matrix is defined the be the first and the last row, together with the first and the last column. The corners of a matrix are the entries (i = 1, j = 1), (i = 1, j = n), (i = n, j = 1) and (i = n, j = n). - _Simone Severini_, Oct 17 2004
%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 520.
%H Reinhard Zumkeller, <a href="/A052849/b052849.txt">Table of n, a(n) for n = 0..400</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=490">Encyclopedia of Combinatorial Structures 490</a>.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=817">Encyclopedia of Combinatorial Structures 817</a>.
%H Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Talman/talman2.html">The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients</a>, J. Int. Seq. (2023) Art. 23.4.3. See p. 11.
%H T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">Avoiding 2-letter signed patterns</a>, arXiv:math/0207204 [math.CO], 2002.
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%F a(n) = T(n, 2) for n>1, where T is defined as in A080046.
%F D-finite with recurrence: {a(0) = 0, a(1) = 2, (-1 - n)*a(n+1) + a(n+2)=0}.
%F E.g.f.: 2*x/(1-x).
%F a(n) = A090802(n, n - 1) for n > 0. - _Ross La Haye_, Sep 26 2005
%F For n >= 1, a(n) = (n+3)!*Sum_{k=0..n+2} (-1)^k*binomial(2, k)/(n + 3 - k). - _Milan Janjic_, Dec 14 2008
%F G.f.: 2/Q(0) - 2, where Q(k) = 1 - x*(k + 1)/(1 - x*(k + 1)/Q(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Apr 01 2013
%F G.f.: -2 + 2/Q(0), where Q(k) = 1 + k*x - x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 01 2013
%F G.f.: W(0) - 2 , where W(k) = 1 + 1/( 1 - x*(k+1)/( x*(k+1) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 21 2013
%F a(n) = A245334(n, n-1), n > 0. - _Reinhard Zumkeller_, Aug 31 2014
%F From _Amiram Eldar_, Jan 15 2023: (Start)
%F Sum_{n>=1} 1/a(n) = (e-1)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (e-1)/(2*e). (End)
%p spec := [S,{B=Cycle(Z),C=Cycle(Z),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t Join[{0}, 2Range[20]!] (* _Harvey P. Dale_, Jul 13 2013 *)
%o (PARI) a(n)=if(n<1,0,n!*2)
%o (Haskell)
%o a052849 n = if n == 0 then 0 else 2 * a000142 n
%o a052849_list = 0 : fs where fs = 2 : zipWith (*) [2..] fs
%o -- _Reinhard Zumkeller_, Aug 31 2014
%o (Magma) [0] cat [2*Factorial(n-1): n in [2..25]]; // _Vincenzo Librandi_, Nov 03 2014
%Y Essentially the same sequence as A098558.
%Y Cf. A062119, A080046, A245334, A000142.
%Y Row 3 of A276955 (from term a(2)=4 onward).
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _Ross La Haye_, Sep 26 2005
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