A111713 Number of reduced tree pairs of n-carets.

0, 1, 2, 14, 108, 930, 8700, 86598, 904176, 9804516, 109624536, 1257136130, 14726063264, 175650153588, 2128038439176, 26133761328150, 324786698542440, 4079191750094776, 51716838331485472, 661227615895716180, 8518677674587163584

Offset: 0

Author

Murray Elder, May 04 2007

Keywords

• nonn

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.