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A303843 The number of unlabeled trees with n nodes rooted at 3 indistinguishable roots.

%I A303843 #17 Mar 16 2025 23:53:16
%S A303843 0,0,1,4,15,51,175,573,1866,5978,19000,59859,187503,584012,1811212,
%T A303843 5595239,17228943,52898764,162013452,495100454,1510029296,4597430832,
%U A303843 13975327501,42422033217,128606150706,389423872694,1177925775148,3559477190797,10746362772325
%N A303843 The number of unlabeled trees with n nodes rooted at 3 indistinguishable roots.
%C A303843 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
%H A303843 Andrew Howroyd, <a href="/A303843/b303843.txt">Table of n, a(n) for n = 1..500</a>
%F A303843 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
%e A303843 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.
%t A303843 m = 30; T[_] = 0;
%t A303843 Do[T[x_] = x Exp[Sum[T[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];
%t A303843 g[x_] = T[x]/(1 - T[x]) + O[x]^m // Normal;
%t A303843 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_ *)
%o A303843 (PARI) \\ here TreeGf is gf of A000081
%o A303843 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)}
%o A303843 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}
%o A303843 concat([0,0], Vec(seq(30))) \\ _Andrew Howroyd_, May 03 2018
%Y A303843 4th column of A294783.
%Y A303843 Cf. A000081 (1-rooted), A303833 (2-rooted).
%K A303843 nonn
%O A303843 1,4
%A A303843 _R. J. Mathar_, May 01 2018
%E A303843 Terms a(11) and beyond from _Andrew Howroyd_, May 03 2018