A280920 - OEIS

%I #9 Mar 02 2017 02:39:07

%S 0,0,0,0,0,720,4320,30960,256320,2399760,25022880,287250480,

%T 3597143040,48773612880,711607724640,11113078385520,184925331414720,

%U 3265974496290960,61006644910213920,1201583921745846960,24885771463659934080,540624959563046320080,12291921453805577987040

%N Seventh column of Euler's difference table in A068106.

%C For n >= 7, this is the number of permutations of [n] that avoid substrings j(j+6), 1 <= j <= n-6.

%H Indranil Ghosh, <a href="/A280920/b280920.txt">Table of n, a(n) for n = 1..400</a>

%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>=7: a(n) = Sum_{j=0..n-6} (-1)^j*binomial(n-6,j)*(n-j)!.

%F Note a(n)/n! ~ 1/e.

%e a(10)=2399760 since there are 2399760 permutations in S10 that avoid substrings {17,28,39,4(10)}.

%t Table[Sum[(-1)^j*Binomial[n-6,j]*(n-j)!,{j,0,n-6}],{n,1,23}] (* _Indranil Ghosh_, Feb 26 2017 *)

%o (Python)

%o f=math.factorial

%o def C(n,r):return f(n)/f(r)/f(n-r)

%o def A280920(n):

%o ....s=0

%o ....for j in range(0,n-5):

%o ........s+=(-1)**j*C(n-6,j)*f(n-j)

%o ....return s # _Indranil Ghosh_, Feb 26 2017

%o (PARI) a(n) = sum(j=0, n-6, (-1)^j*binomial(n-6,j)*(n-j)!); \\ _Michel Marcus_, Feb 26 2017

%Y Also 720 times A176732.

%Y Cf. A068106.

%K nonn

%O 1,6

%A _Enrique Navarrete_, Jan 10 2017