A277609 Fourth column of Euler's difference table in A068106. It is 6 times the sequence A000261.

0, 0, 6, 18, 78, 426, 2790, 21234, 183822, 1781802, 19104774, 224406930, 2864826126, 39486808938, 584328412518, 9238767895026, 155416555683150, 2771424197143914, 52216883883837702, 1036463580947218962, 21616958644969620174, 472612476001411964970, 10808196686285486012646

Offset: 1

Author

Enrique Navarrete, Oct 23 2016

Keywords

• nonn

Comments

For n >= 4, this is the number of permutations that avoid substrings j(j+3), 1 <= j <= n-3.

For n>=4, the number of circular permutations (in cycle notation) on [n+1] that avoid substrings (j,j+4), 1<=j<=n-3. For example, for n=4, there are 18 circular permutations in S5 that avoid the substring {15}. Note that each of these circular permutations represent 5 permutations in one-line notation (see link 2017). - Enrique Navarrete, Feb 22 2017

Examples

			a(5) = 78 since there are 78 permutations in S5 that avoid the substrings {14,25}.
		

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.
• Enrique Navarrete, Forbidden Substrings in Circular K-Successions, arXiv:1702.02637 [math.CO], 2017.

Crossrefs

Cf. A068106, A000261.

Programs

Mathematica

Table[Sum[(-1)^j*Binomial[n - 3, j] (n - j)!, {j, 0, n - 3}], {n, 23}] (* Michael De Vlieger, Oct 27 2016 *)
Flatten[{0, 0, Table[n!*Hypergeometric1F1[3-n, -n, -1], {n, 3, 20}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

PARI

a(n) = sum(j=0, n-3, (-1)^j*binomial(n-3,j)*(n-j)!); \\ Michel Marcus, Oct 29 2016

Formula

For n>=4: a(n) = Sum_{j=0..n-3} (-1)^j*binomial(n-3,j)*(n-j)!.

a(n) ~ exp(-1) * n!. - Vaclav Kotesovec, Oct 28 2016