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A378672 Numbers of uniquely embeddable trees on n vertices.

%I A378672 #14 Jun 08 2025 17:33:02
%S A378672 1,1,1,2,3,6,10,19,31,57,95,161,262,435,683,1081,1665,2545,3800,5658,
%T A378672 8232,11935,17012,24074,33606,46698,63994,87281,117797,158121,210240,
%U A378672 278389,365267,477416,619278,799962,1026370,1312419,1667131,2111086
%N A378672 Numbers of uniquely embeddable trees on n vertices.
%H A378672 Andrew Howroyd, <a href="/A378672/b378672.txt">Table of n, a(n) for n = 1..1000</a>
%H A378672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UniquelyEmbeddableGraph.html">Uniquely Embeddable Graph</a>.
%H A378672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Tree.html">Tree</a>.
%F A378672 a(n) = A000055(n) - A378673(n).
%F A378672 G.f.: x + ((1 - x)*g(x)^2 + (1 + x)*g(x^2))/2 + x*(g(x)^3 - 3*g(x)*g(x^2) + 2*g(x^3))/6, where g(x) is the g.f. of A003238. - _Andrew Howroyd_, Jun 08 2025
%o A378672 (PARI) \\ G(n) is A003238 as g.f.
%o A378672 G(n) = {my(v=vector(n)); v[1]=1; for(i=2, n, v[i]=sumdiv(i-1, d, v[d])); x*Ser(v)}
%o A378672 seq(n) = {my(g=G(n-1)); Vec(x + ((1 - x)*g^2 + (1 + x)*subst(g,x,x^2))/2 + x*(g^3 - 3*g*subst(g,x,x^2) + 2*subst(g,x,x^3))/6)} \\ _Andrew Howroyd_, Jun 08 2025
%Y A378672 Cf. A378673 (not uniquely embeddable trees).
%Y A378672 Cf. A372853 (uniquely embeddable planar connected graphs).
%Y A378672 Cf. A000055 (trees), A003238.
%K A378672 nonn
%O A378672 1,4
%A A378672 _Eric W. Weisstein_, Dec 03 2024
%E A378672 a(13) onwards from _Andrew Howroyd_, Jun 08 2025