OFFSET
0,3
COMMENTS
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.
LINKS
S. Cleary, M. Elder, A. Rechnitzer and J. Taback, Random subgroups of Thompson's group F, arxiv:0711.1343 (2007)
S. Cleary, M. Elder, A. Rechnitzer, J. Taback, Random subgroups of Thompson's group F, Groups, Geom. Dynam. 4 (1) (2010) 91-126
Benjamin M. Woodruff, Statistical Properties of Thompson's Group and Random Pseudo Manifolds
Wikipedia, Thompson groups
FORMULA
a(n) = Sum_{k=1..n} (-1)^(k+n) * binomial(k+1,n-k) * ( binomial(2*k,k)/(k+1) )^2.
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)*(d/dq)A(q) + q^2*(-1+q)*(2*q-1)*(16*q^2-16*q+1)*(d^2/dq^2)A(q) - 4*q*(-1+q)*(2*q-1)^3.
CROSSREFS
KEYWORD
nonn
AUTHOR
Murray Elder, May 04 2007
STATUS
approved