A378672 Numbers of uniquely embeddable trees on n vertices.

1, 1, 1, 2, 3, 6, 10, 19, 31, 57, 95, 161, 262, 435, 683, 1081, 1665, 2545, 3800, 5658, 8232, 11935, 17012, 24074, 33606, 46698, 63994, 87281, 117797, 158121, 210240, 278389, 365267, 477416, 619278, 799962, 1026370, 1312419, 1667131, 2111086

Offset: 1

Author

Eric W. Weisstein, Dec 03 2024

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Uniquely Embeddable Graph.
• Eric Weisstein's World of Mathematics, Tree.

Crossrefs

Cf. A378673 (not uniquely embeddable trees).

Cf. A372853 (uniquely embeddable planar connected graphs).

Cf. A000055 (trees), A003238.

Programs

PARI

\\ G(n) is A003238 as g.f.
G(n) = {my(v=vector(n)); v[1]=1; for(i=2, n, v[i]=sumdiv(i-1, d, v[d])); x*Ser(v)}
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

Formula

a(n) = A000055(n) - A378673(n).

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

Extensions

a(13) onwards from Andrew Howroyd, Jun 08 2025