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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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

4th column of A294783.

Cf. A000081 (1-rooted), A303833 (2-rooted).

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

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Last modified June 17 04:38 EDT 2021. Contains 345080 sequences. (Running on oeis4.)