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%I #12 Jan 27 2022 12:56:29
%S 2,12,336,17760,1543680,199019520,35611269120,8437755432960,
%T 2556188496691200,963558923688345600,442230750973683302400,
%U 242766600433441859174400,157060798435284559803187200
%N Number of permutations of {1,2,...,2n} , say x(1),x(2), ... , x(2n) , such that x(i) + x(i+1) is not equal to 2n-1 for all i, 1<=i<=2n-1.
%H Nathan Reading, <a href="http://www4.ncsu.edu/~nreadin/524/assignments.html">Assignments</a>
%H StackExchange, <a href="http://math.stackexchange.com/questions/6220/can-this-sum-be-simplified">Can this be simplified?</a>
%F a(n) = Sum_{k=0}^{n-1} binomial(n -1,k) (-2)^k (2n - k)!.
%F Conjecture: D-finite with recurrence (-n+1)*a(n) +2*(2*n-1)*(n^2-n-1)*a(n-1) +4*n^2*(n-2)*a(n-2)=0. - _R. J. Mathar_, Jan 27 2022
%t f[n_] := Sum[(-2)^k (2 n - k)! Binomial[n - 1, k], {k, 0, n - 1}]; Array[f, 13] (* _Robert G. Wilson v_, Oct 16 2010 *)
%o (Other) SAS datastep: data _null_; do n = 1 to 7; a = 0; do _n_ = 0 to n-1; a = a + (-2)**_n_ * comb(n-1,_n_)*fact(2*n-_n_); end; output; put "a(" n ")=" a; end; run;
%K nonn
%O 1,1
%A Arin Chaudhuri (arin.chaudhuri(AT)gmail.com), Oct 06 2010
%E a(8) and onward from _Robert G. Wilson v_, Oct 16 2010
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