1,4

It is not known if these trees have the asymptotic form C rho^{-n} n^{-3/2}, whereas the identity binary trees, A063895, do, see the Jason P. Bell et al. reference.

Jason P. Bell, Stanley N. Burris and Karen A. Yeats, Counting Rooted Trees: The Universal Law t(n) ~ C rho^{-n} n^{-3/2}

G.f. satisfies A(x) = x(1 + A(x) + A(x)^2/2-A(x^2)/2 + A(x)^3/6-A(x)A(x^2)/2+A(x^3)/3 + A(x)^4/24-A(x)^2A(x^2)/4+A(x)A(x^3)/3+A(x^2)^2/8-A(x^4)/4), that is A(x) = x(1+Set_{<=4}(A)(x))

A:= proc(n) option remember; local T; if n<=1 then x else T:= unapply (A(n-1), x); convert (series (x* (1+T(x)+ T(x)^2/2- T(x^2)/2+ T(x)^3/6- T(x)*T(x^2)/2+ T(x^3)/3+ T(x)^4/24- T(x)^2* T(x^2)/4+ T(x)* T(x^3)/3+ T(x^2)^2/8- T(x^4)/4), x, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n): seq (a(n), n=1..32); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 22 2008]

(C) #include <ginac/ginac.h> using namespace GiNaC; int main(){ int i, order=40; symbol x("x"); ex T; for (i=0; i<order; i++) T = (x+x*(T + pow(T, 2)/2 - T.subs(x==pow(x, 2))/2 + pow(T, 3)/6 - T*T.subs(x==pow(x, 2))/2 + T.subs(x==pow(x, 3))/3 + pow(T, 4)/24 - pow(T, 2)*T.subs(x==pow(x, 2))/4 + T*T.subs(x==pow(x, 3))/3 + pow(T.subs(x==pow(x, 2)), 2)/8 - T.subs(x==pow(x, 4))/4)).series(x, i+3); for (i=1; i<=order; i++) std::cout << T.coeff(x, i) << ", "; }

Cf. A004111, A063895, A116379.

Sequence in context: A005829 A038087 A116379 this_sequence A004111 A032235 A052523

Adjacent sequences: A116377 A116378 A116379 this_sequence A116381 A116382 A116383

nonn

Karen Yeats (kayeats(AT)bu.edu), Feb 06 2006