OFFSET

1,4

COMMENTS

A unique path exists between any two of the roots. These will intersect at a single vertex which might coincide with one of the original roots. This intersecting vertex can be chosen as a root to which the other trees are attached. - Andrew Howroyd, May 03 2018

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..500

FORMULA

G.f.: g(x)*(g(x)^3 + 3*g(x)*g(x^2) + 2*g(x^3) + 3*g(x)^2 + 3*g(x^2))/(6*(1 + g(x)) where g(x)=T(x)/(1-T(x)) and T(x) is the g.f. of A000081. - Andrew Howroyd, May 03 2018

EXAMPLE

a(3)=1 (all nodes are roots). a(4)=4: the linear tree has the non-root node either at a leaf or not, and the star tree has the non-root node either at the center or at a leaf.

MATHEMATICA

m = 30; T[_] = 0;

Do[T[x_] = x Exp[Sum[T[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];

g[x_] = T[x]/(1 - T[x]) + O[x]^m // Normal;

g[x]((g[x]^3 + 3g[x]g[x^2] + 2g[x^3] + 3g[x]^2 + 3g[x^2])/(6(1 + g[x]))) + O[x]^m // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 16 2020, after Andrew Howroyd *)

PROG

(PARI) \\ here TreeGf is gf of A000081

TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}

seq(n) = {my(T=TreeGf(n)); my(g=T/(1-T)); T*(g^3 + 3*subst(g, x, x^2)*g + 2*subst(g, x, x^3) + 3*g^2 + 3*subst(g, x, x^2))/6}

concat([0, 0], Vec(seq(30))) \\ Andrew Howroyd, May 03 2018

CROSSREFS

KEYWORD

nonn

AUTHOR

R. J. Mathar, May 01 2018

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, May 03 2018

STATUS

approved