A277563 Fifth column of Euler's difference table in A068106.

0, 0, 0, 24, 96, 504, 3216, 24024, 205056, 1965624, 20886576, 243511704, 3089233056, 42351635064, 623815221456, 9823096307544, 164655323578176, 2926840752827064, 54988308080981616, 1088680464831056664, 22653422225916839136, 494229434646381585144, 11280809162286897977616

Offset: 1

Author

Enrique Navarrete, Dec 03 2016

Keywords

• nonn

Comments

This is 24 times the sequence A001909.

For n >= 5, this is the number of permutations that avoid substrings j(j+4), 1 <= j <= n-4.

For n>=5, the number of circular permutations (in cycle notation) on [n+1] that avoid substrings (j,j+5), 1<=j<=n-4. For example, for n=5, there are 96 circular permutations in S6 that avoid the substring {16}. Note that each of these circular permutations represent 6 permutations in one-line notation (see link 2017). - Enrique Navarrete, Feb 22 2017

Examples

			a(6) = 504 since there are 504 permutations in S6 that avoid the substrings {15,26}.
		

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. A001909, A068106.

Programs

Mathematica

Array[Sum[(-1)^j*Binomial[# - 4, j] (# - j)!, {j, 0, # - 4} ] &, 23] (* Michael De Vlieger, Dec 06 2016 *)

Formula

For n>=5: a(n) = Sum_{j=0..n-4} (-1)^j*binomial(n-4,j)*(n-j)!.

a(n) ~ n!/e.