0,3

The number of ordered pairs of rooted binary trees such that each tree has n ``carets'' and the pair is ``reduced''. A ``caret'' is a vertex with two (downward) edges. Number the leaves of each tree from left to right (infix order). A tree-pair is ``reduced'' if i,i+1 is not the label of a caret in both trees for any i.

The elements of Thompson's group F can be represented uniquely as a reduced tree pair. a(n) is asymptotic to (12/ Pi / mu) * mu^n/n^3*(1 + O(1/n)) and so the corresponding g.f. cannot be algebraic.

S. Cleary, M. Elder, A. Rechnitzer and J. Taback, Asymptotic properties of Thompson's group F, to appear.

Author?, Title?

Wikipedia, Thompson groups

a(n)=sum((-1)^(k+n) * binomial(k+1,n-k) * ( binomial(2*k,k)/(k+1) )^2,k=1..n)

0 = (16*q^3-6*q^2-6*q+1)*A(q)+q*(4*q-3)*(8*q^3-18*q^2+12*q-1)*diff(A(q),q)+q^2*(-1+q)*(2*q-1)*(16*q^2-16*q+1)*diff(A(q), q, q)-4*q*(-1+q)*(2*q-1)^3

Sequence in context: A108436 A088754 A103945 this_sequence A144278 A092639 A155728

Adjacent sequences: A111710 A111711 A111712 this_sequence A111714 A111715 A111716

nonn

Murray Elder (murrayelder(AT)gmail.com), May 04 2007