A082938 Number of solid 2-trees with 2n+1 edges.

1, 1, 1, 2, 5, 13, 49, 201, 940, 4643, 24037, 127859, 696365, 3858759, 21704863, 123619126, 711787259, 4137614454, 24256010068, 143271593982, 852001881614, 5097719884665, 30670572676389, 185466705697057

Offset: 0

Author

N. J. A. Sloane, May 26 2003

Keywords

• nonn

Comments

Also, the number of noncrossing partitions up to rotation and reflection composed of n blocks of size 3. - Andrew Howroyd, May 03 2018

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141.

Crossrefs

Column k=3 of A303929.

Cf. A047749, A054423.

Programs

Mathematica

u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #] &] + DivisorSum[GCD[n-1, k], EulerPhi[#]*Binomial[n*k/#, (n-1)/#] &])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
a[n_] := T[n, 3];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd and A303929 *)

Formula

a(n) = (A047749(n)+A054423(n))/2. - Vladeta Jovovic, Sep 11 2004

a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Jun 01 2022

Extensions

More terms from Vladeta Jovovic, Sep 11 2004