OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
M. D. Atkinson, M. H. Albert, R. E. L. Aldred, H. P. van Ditmarsch, C. C. Handley, D. A. Holton, D. J. McCaughan, C. Monteith, Cyclically closed pattern classes of permutations, Australasian J. Combinatorics 38 (2007), 87-100.
R. Brignall, S. Huczynska, V. Vatter, Simple permutations and algebraic generating functions, J. Combinatorial Theory, Series A 115 (2008), 423-441.
FORMULA
G.f.: (1-4*x+4*x^2-4*x^3-(1-2*x)*sqrt(1-4*x))/(2*x*(1-x)^2*sqrt(1-4*x)).
a(n) = n(C(n) - C(n-1) - ... - C(1)), where C(n) denotes the n-th Catalan number.
a(n) ~ 2^(2*n+1)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence -3*(n+1)*(n-3)*a(n) +n*(17*n-43)*a(n-1) +2*(-11*n^2+35*n-30)*a(n-2) +4*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 19 2022
D-finite with recurrence (n-1)*(n-3)*(n+1)*a(n) -n*(5*n^2-16*n+9)*a(n-1) +2*n*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 19 2022
EXAMPLE
a(5)=100 because 100 permutations of length 5 have at least one cyclic rotation which avoids 132.
MATHEMATICA
Rest[CoefficientList[Series[(1-4*x+4*x^2-4*x^3-(1-2*x)*Sqrt[1-4*x]) / (2*x*(1-x)^2*Sqrt[1-4*x]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) x='x+O('x^50); Vec((1-4*x+4*x^2-4*x^3-(1-2*x)*sqrt(1-4*x))/(2*x*(1-x)^2*sqrt(1-4*x))) \\ G. C. Greubel, Mar 21 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vincent Vatter, Jun 17 2008
STATUS
approved