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

1,4

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.

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

Prog

(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

Crossrefs

Cf. A378673 (not uniquely embeddable trees).

Cf. A372853 (uniquely embeddable planar connected graphs).

Cf. A000055 (trees), A003238.

Sequence in context: A089985 A191518 A217382 * A244742 A007473 A014595

Adjacent sequences: A378669 A378670 A378671 * A378673 A378674 A378675

Keyword

nonn

Author

Eric W. Weisstein, Dec 03 2024

Extensions

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

Status

approved