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A116380
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Number of quaternary rooted identity (distinct subtrees) trees with n nodes.
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3
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1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6298, 14419, 33183, 76760, 178327, 415960, 973693, 2286781, 5386573, 12723097, 30127465, 71506140, 170081575, 405359177, 967899981, 2315131955, 5546597838, 13308818691, 31979667219, 76947325788
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OFFSET
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1,4
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COMMENTS
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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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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 + A(x)^4/24-A(x)^2A(x^2)/4+A(x)A(x^3)/3+A(x^2)^2/8-A(x^4)/4), that is A(x) = x(1+Set_{<=4}(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+ T(x)^4/24- T(x)^2* T(x^2)/4+ T(x)* T(x^3)/3+ T(x^2)^2/8- T(x^4)/4), 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
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MATHEMATICA
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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+T[y]^4/24-T[y]^2*T[y^2]/4+T[y]*T[y^3]/3+T[y^2]^2/8-T[y^4]/4), {y, 0, n+1}]] /. y -> x]; a[n_] := Coefficient[A[n], x, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 13 2014, after Maple *)
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PROG
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(C) #include <ginac/ginac.h> using namespace GiNaC; int main(){ int i, order=40; symbol x("x"); ex T; 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 + pow(T, 4)/24 - pow(T, 2)*T.subs(x==pow(x, 2))/4 + T*T.subs(x==pow(x, 3))/3 + pow(T.subs(x==pow(x, 2)), 2)/8 - T.subs(x==pow(x, 4))/4)).series(x, i+3); for (i=1; i<=order; i++) std::cout << T.coeff(x, i) << ", "; }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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