OFFSET
0,3
COMMENTS
Permutations of 12...n such that exactly one of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
REFERENCES
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200
Sergey Kitaev, Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.
FORMULA
Coefficient of t^1 in S[n](t) defined in A002464.
(3-n)*a(n) +(n+1)*(n-3)*a(n-1) -(n^2-4*n+5)*a(n-2) -(n-1)*(n-5)*a(n-3) +(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 06 2013
a(n) ~ 2*sqrt(2*Pi)*n!/exp(2) = 0.678470495... * n!. - Vaclav Kotesovec, Aug 10 2013
MAPLE
S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
[n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
-(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
end:
a:= n-> coeff(S(n), t, 1):
seq(a(n), n=0..30); # Alois P. Heinz, Dec 21 2012
MATHEMATICA
S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(n+1-t)*S[n-1]-(1-t)*(n-2+3*t)*S[n-2]-(1-t)^2*(n-5+t)*S[n-3]+(1-t)^3*(n-3)*S[n-4]]]; a[n_] := Coefficient[S[n], t, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 19 2003
STATUS
approved