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%I #8 Apr 03 2017 10:34:46
%S 0,0,0,0,0,0,0,40320,322560,2943360,30078720,339696000,4196666880,
%T 56255149440,812752093440,12585067447680,207863095910400,
%U 3646938237505920,67723519234210560,1326863186062565760,27349945952061841920,591598086412112035200
%N Ninth column of Euler's difference table in A068106.
%C For n >= 9, this is the number of permutations of [n] that avoid substrings j(j+8), 1 <= j <= n-8.
%H Enrique Navarrete, <a href="http://arxiv.org/abs/1610.06217">Generalized K-Shift Forbidden Substrings in Permutations</a>, arXiv:1610.06217 [math.CO], 2016.
%F For n>=9: a(n) = Sum_{j=0..n-8} (-1)^j*binomial(n-8,j)*(n-j)!.
%F Note a(n)/n! ~ 1/e.
%e a(12)=339696000 since this is the number of permutations in S12 that avoid substrings {19,2(10),3(11),4(12)}.
%t With[{k = 9}, ConstantArray[0, k - 2]~Join~Table[Sum[(-1)^j*Binomial[n - (k - 1), j] (n - j)!, {j, 0, n - (k - 1)}], {n, k - 1, k + 12}]] (* _Michael De Vlieger_, Mar 26 2017 *)
%Y Also 40320 times A176734.
%Y Cf. A068106.
%K nonn
%O 1,8
%A _Enrique Navarrete_, Mar 22 2017