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A280920 Seventh column of Euler's difference table in A068106.

%I A280920 #11 May 22 2025 10:21:45
%S A280920 0,0,0,0,0,720,4320,30960,256320,2399760,25022880,287250480,
%T A280920 3597143040,48773612880,711607724640,11113078385520,184925331414720,
%U A280920 3265974496290960,61006644910213920,1201583921745846960,24885771463659934080,540624959563046320080,12291921453805577987040
%N A280920 Seventh column of Euler's difference table in A068106.
%C A280920 For n >= 7, this is the number of permutations of [n] that avoid substrings j(j+6), 1 <= j <= n-6.
%H A280920 Indranil Ghosh, <a href="/A280920/b280920.txt">Table of n, a(n) for n = 1..400</a>
%H A280920 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 A280920 For n>=7: a(n) = Sum_{j=0..n-6} (-1)^j*binomial(n-6,j)*(n-j)!.
%F A280920 Note a(n)/n! ~ 1/e.
%e A280920 a(10)=2399760 since there are 2399760 permutations in S10 that avoid substrings {17,28,39,4(10)}.
%t A280920 Table[Sum[(-1)^j*Binomial[n-6,j]*(n-j)!,{j,0,n-6}],{n,1,23}] (* _Indranil Ghosh_, Feb 26 2017 *)
%o A280920 (Python)
%o A280920 f=math.factorial
%o A280920 def C(n,r):return f(n)/f(r)/f(n-r)
%o A280920 def A280920(n):
%o A280920     s=0
%o A280920     for j in range(0,n-5):
%o A280920         s+=(-1)**j*C(n-6,j)*f(n-j)
%o A280920     return s # _Indranil Ghosh_, Feb 26 2017
%o A280920 (PARI) a(n) = sum(j=0, n-6, (-1)^j*binomial(n-6,j)*(n-j)!); \\ _Michel Marcus_, Feb 26 2017
%Y A280920 Also 720 times A176732.
%Y A280920 Cf. A068106.
%K A280920 nonn
%O A280920 1,6
%A A280920 _Enrique Navarrete_, Jan 10 2017