|
| |
|
|
A111713
|
|
Number of reduced tree pairs of n-carets.
|
|
0
| |
|
|
0, 1, 2, 14, 108, 930, 8700, 86598, 904176, 9804516, 109624536, 1257136130, 14726063264, 175650153588, 2128038439176, 26133761328150, 324786698542440, 4079191750094776, 51716838331485472, 661227615895716180, 8518677674587163584
(list; graph; refs; listen; history; internal format)
|
|
|
|
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.
|
|
|
REFERENCES
| S. Cleary, M. Elder, A. Rechnitzer and J. Taback, Asymptotic properties of Thompson's group F, to appear.
|
|
|
LINKS
| Author?, Title?
Wikipedia, Thompson groups
|
|
|
FORMULA
| 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
|
|
|
CROSSREFS
| Sequence in context: A108436 A088754 A103945 * A144278 A199649 A192406
Adjacent sequences: A111710 A111711 A111712 * A111714 A111715 A111716
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Murray Elder (murrayelder(AT)gmail.com), May 04 2007
|
| |
|
|
