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

1,4

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

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.

Formula

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

a(n) ~ n!/e.

Example

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

Mathematica

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

Crossrefs

Cf. A001909, A068106.

Sequence in context: A183009 A272871 A319577 * A225790 A042122 A042124

Adjacent sequences: A277560 A277561 A277562 * A277564 A277565 A277566

Keyword

nonn

Author

Enrique Navarrete, Dec 03 2016

Status

approved