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A116379 Number of ternary rooted identity (distinct subtrees) trees with n nodes. 4
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, 27172368314, 64875527649 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Jason P. Bell, Stanley N. Burris and Karen A. Yeats, Counting Rooted Trees: The Universal Law t(n) ~ C rho^{-n} n^{-3/2}

FORMULA

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))

MAPLE

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..40);  # Alois P. Heinz, Aug 22 2008

MATHEMATICA

A[n_] := A[n] = If[n <= 1, x, T[y_] = A[n-1] /. x -> y; Normal[Series[y*(1+T[y]+T[y]^2/2-T[y^2]/2+T[y]^3/6-T[y]*T[y^2]/2+T[y^3]/3), {y, 0, n+1}]] /. y -> x] ; a[n_] := Coefficient[A[n], x, n]; Table[a[n], {n, 1, 40}] (* Jean-Fran├žois Alcover, Feb 13 2014, after Maple *)

PROG

(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) << ", "; }

CROSSREFS

Cf. A004111, A063895, A116380.

Sequence in context: A032218 A005829 A038087 * A116380 A004111 A032235

Adjacent sequences:  A116376 A116377 A116378 * A116380 A116381 A116382

KEYWORD

nonn

AUTHOR

Karen Yeats (kayeats(AT)bu.edu), Feb 06 2006

STATUS

approved

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Last modified January 17 23:37 EST 2020. Contains 330995 sequences. (Running on oeis4.)