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Search: id:A116379
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| A116379 |
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Number of ternary rooted identity (distinct subtrees) trees with n nodes. |
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+0
2
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| 1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 245, 542, 1205, 2707, 6113, 13907, 31780, 73010, 168399, 389991, 906231, 2112742, 4939689, 11580640, 27216387, 64110091, 151334814, 357938832, 848153045, 2013190671, 4786210412, 11396004660
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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It is not known if these trees have the asymptotic form C rho^{-n} n^{-3/2}, whereas the identity binary trees, A063895, do, see the Jason P. Bell et al. reference.
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LINKS
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Jason P. Bell, Stanley N. Burris and Karen A. Yeats, Counting Rooted Trees: The Universal Law t(n) ~ C rho^{-n} n^{-3/2}
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FORMULA
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G.f. satisfies A(x) = x(1+A(x)+A(x)^2/2-A(x^2)/2+A(x)^3/6-A(x)A(x^2)/2+A(x^3)/3), that is A(x) = x(1+Set_{<=3}(A)(x))
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MAPLE
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A:= proc(n) option remember; local T; if n<=1 then x else T := unapply (A(n-1), x); convert (series (x* (1+T(x)+ T(x)^2/2- T(x^2)/2+ T(x)^3/6- T(x)*T(x^2)/2+ T(x^3)/3), x, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n): seq (a(n), n=1..32); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 22 2008]
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PROGRAM
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(C) #include <ginac/ginac.h> using namespace GiNaC; int main(){ int i, order = 40; symbol x("x"); ex T = x; for (i=0; i<order; i++) T = (x+x*(T + pow(T, 2)/2 - T.subs(x==pow(x, 2))/2 + pow(T, 3)/6 - T*T.subs(x==pow(x, 2))/2 + T.subs(x==pow(x, 3))/3)).series(x, i+2); for (i=1; i<=order; i++) std::cout << T.coeff(x, i) << ", "; }
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CROSSREFS
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Cf. A004111, A063895, A116380.
Sequence in context: A032218 A005829 A038087 this_sequence A116380 A004111 A032235
Adjacent sequences: A116376 A116377 A116378 this_sequence A116380 A116381 A116382
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KEYWORD
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nonn
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AUTHOR
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Karen Yeats (kayeats(AT)bu.edu), Feb 06 2006
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