0,3

Permutations on n letters without double falls. A permutation w has a double fall at k if w(k) > w(k+1) > w(k+2) and has an initial fall if w(1) > w(2).

Hankel transform is A055209. [From Paul Barry (pbarry(AT)wit.ie), Jan 12 2009]

M. Aigner, Catalan and other numbers: a recurrent theme, Algebraic combinatorics and computer science, 347-390, Springer Italia, Milan, 2001.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.17).

S. Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns

G.f.: Sum(a_n x^n / n!, n=0, inf) = 1 / sum(x^(3i)/(3i)! - x^(3i+1)/(3i+1)!, i=1, inf).

Equivalently, g.f. = exp(x/2) * r / sin(r*x + (2/3)Pi) where r = sqrt(3)/2. This has simple poles at (3m+1)*x0 where x0 = Pi/sqrt(6.75) = 1.2092 approximately and m is an arbitrary integer. This yields the asymptotic expansion a_n / n! ~ x0^(-n-1) sum((-1)^m * E^(3*m+1) / (3*m+1)^(n+1)) where E = exp(x0/2) = 1.8305+ and m ranges over all integers. - Noam D. Elkies, Nov 15, 2001

E.g.f.: Sqrt[3] Exp[x/2] /( Sqrt[3] Cos[Sqrt[3]/2 x] - Sin[Sqrt[3]/2 x] ) Recurrence: a(n+1) = Sum[ binomial[n, k] a(k) b(n-k), {k, 0, n} ] where b(n) = number of n-permutations without double falls and without initial falls. - Emanuele Munarini (munarini(AT)mate.polimi.it), Feb 28 2003

O.g.f.: A(x) = 1/(1-x-x^2/(1-2*x-4*x^2/(1-3*x-9*x^2/(1-... -n*x-n^2*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 17 2006

For n = 3: 123, 132, 213, 231, 312.

Table[Simplify[ n! SeriesCoefficient[ Series[ Sqrt[3] Exp[x/2]/(Sqrt[3] Cos[Sqrt[3]/2 x] - Sin[Sqrt[3]/2 x]), {x, 0, n}], n] ], {n, 0, 40}]

Cf. A065429, A080635.

Sequence in context: A027361 A101971 A162037 this_sequence A139402 A143382 A057219

Adjacent sequences: A049771 A049772 A049773 this_sequence A049775 A049776 A049777

nonn,nice,easy

Tuwani A. Tshifhumulo (tat(AT)caddy.univen.ac.za)

Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 14 2001