

A303843


The number of unlabeled trees with n nodes rooted at 3 indistinguishable roots.


2



0, 0, 1, 4, 15, 51, 175, 573, 1866, 5978, 19000, 59859, 187503, 584012, 1811212, 5595239, 17228943, 52898764, 162013452, 495100454, 1510029296, 4597430832, 13975327501, 42422033217, 128606150706, 389423872694, 1177925775148, 3559477190797, 10746362772325
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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)/(1T(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 nonroot node either at a leaf or not, and the star tree has the nonroot 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 (* JeanFrançois 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, N1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[nk+1] ) ); x*Ser(A)}
seq(n) = {my(T=TreeGf(n)); my(g=T/(1T)); 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

4th column of A294783.
Cf. A000081 (1rooted), A303833 (2rooted).
Sequence in context: A283276 A196835 A055218 * A107307 A240365 A005492
Adjacent sequences: A303840 A303841 A303842 * A303844 A303845 A303846


KEYWORD

nonn


AUTHOR

R. J. Mathar, May 01 2018


EXTENSIONS

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


STATUS

approved



