0,4

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).

E.g.f. A(x) satisfies A'=1-A+A^2. - Michael Somos, Oct 04 2003

exp(x*(1-y+y^2)*D_y)*f(y)|_{y=0} = f(1-E(-x)) for any function f with a Taylor series. D_y means differentiation with respect to y and E(x) is the e.g.f. given below. For a proof of exp(x*g(y)*D_y)*f(y) = f(F^{-1}(x+F(y))) with the compositional inverse F^{-1} of F(y)=int(1/g(y),y) with F(0)=0 see, e.g., the Datolli et al. reference.

a(n) = number of increasing ordered trees on vertex set {1,2,...,n}, rooted at 1, in which all outdegrees are <=2. - David Callan (callan(AT)stat.wisc.edu), Mar 30 2007

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

G. Datolli, P. L. Ottaviani, A. Torre and L. V\'azquez, Evolution operator equations: integrationn with algebraic and finite differences methods.[...], Rivista del Nuovo Cimento 20,2 (1997) 1-133. eq. (I.2.18).

G.f. : ( 1 + 1/Sqrt[3] Tan[Sqrt[3]/2 x] ) / ( 1 - 1/Sqrt[3] Tan[Sqrt[3]/2 x] ) Recurrece: a(n+1) = Sum[ Binomial[n, k] a(k) a(n-k), {k, 0, n} ] - a(n) + 0^n

E.g.f.: E(x)=(3*cos(1/2*3^(1/2)*x)+(3^(1/2))*sin(1/2*3^(1/2)*x))/(3*cos(1/2*3^(1/2)*x)-(3^(1/2))* sin(1/2*3^(1/2)*x)). See the M. Somos comment. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005.

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

For n = 3 : 123, 132, 231. For n = 4 : 1234, 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.

Table[n! SeriesCoefficient[Series[ (1 + 1/Sqrt[3] Tan[Sqrt[3]/2 x])/(1 - 1/Sqrt[3] Tan[Sqrt[3]/2 x]), {x, 0, n}], n], {n, 0, 40}]

(PARI) a(n)=local(A); if(n<1, n==0, A=O(x); for(k=1, n, A=intformal(1+A+A^2)); n!*polcoeff(A, n))

Cf. A049774.

Sequence in context: A149026 A149027 A121101 this_sequence A130905 A030799 A058105

Adjacent sequences: A080632 A080633 A080634 this_sequence A080636 A080637 A080638

easy,nonn

Emanuele Munarini (munarini(AT)mate.polimi.it), Feb 28 2003