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

0, 0, 0, 0, 0, 720, 4320, 30960, 256320, 2399760, 25022880, 287250480, 3597143040, 48773612880, 711607724640, 11113078385520, 184925331414720, 3265974496290960, 61006644910213920, 1201583921745846960, 24885771463659934080, 540624959563046320080, 12291921453805577987040

Offset: 1

Author

Enrique Navarrete, Jan 10 2017

Keywords

• nonn

Comments

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

Examples

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

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..400
• Enrique Navarrete, Generalized K-Shift Forbidden Substrings in Permutations, arXiv:1610.06217 [math.CO], 2016.

Crossrefs

Also 720 times A176732.

Cf. A068106.

Programs

Mathematica

Table[Sum[(-1)^j*Binomial[n-6,j]*(n-j)!,{j,0,n-6}],{n,1,23}] (* Indranil Ghosh, Feb 26 2017 *)

PARI

a(n) = sum(j=0, n-6, (-1)^j*binomial(n-6,j)*(n-j)!); \\ Michel Marcus, Feb 26 2017

Python

f=math.factorial
def C(n,r):return f(n)/f(r)/f(n-r)
def A280920(n):
    s=0
    for j in range(0,n-5):
        s+=(-1)**j*C(n-6,j)*f(n-j)
    return s # Indranil Ghosh, Feb 26 2017

Formula

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

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