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%I #18 Dec 25 2017 03:49:52
%S 0,1,2,14,108,930,8700,86598,904176,9804516,109624536,1257136130,
%T 14726063264,175650153588,2128038439176,26133761328150,
%U 324786698542440,4079191750094776,51716838331485472,661227615895716180,8518677674587163584
%N Number of reduced tree pairs of n-carets.
%C 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.
%C 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.
%H S. Cleary, M. Elder, A. Rechnitzer and J. Taback, <a href="http://arxiv.org/abs/0711.1343">Random subgroups of Thompson's group F</a>, arxiv:0711.1343 (2007)
%H S. Cleary, M. Elder, A. Rechnitzer, J. Taback, <a href="http://dx.doi.org/10.4171/GGD/76">Random subgroups of Thompson's group F</a>, Groups, Geom. Dynam. 4 (1) (2010) 91-126
%H Benjamin M. Woodruff, <a href="http://contentdm.lib.byu.edu/cdm4/item_viewer.php?CISOROOT=/ETD&CISOPTR=388&CISOBOX=1&REC=3">Statistical Properties of Thompson's Group and Random Pseudo Manifolds</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Thompson_groups">Thompson groups</a>
%F a(n) = Sum_{k=1..n} (-1)^(k+n) * binomial(k+1,n-k) * ( binomial(2*k,k)/(k+1) )^2.
%F 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.
%K nonn
%O 0,3
%A _Murray Elder_, May 04 2007
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