A088325 Piet Hut's "coat-hanger" sequence: unlabeled forests of rooted trees with n edges, where there can be any number of components, the outdegree of each node is <= 2 and the symmetric group acts on the components.

1, 1, 2, 4, 8, 16, 34, 71, 153, 332, 730, 1617, 3620, 8148, 18473, 42097, 96420, 221770, 512133, 1186712, 2758707, 6431395, 15033320, 35224825, 82720273, 194655030, 458931973, 1083926784, 2564305754, 6075896220, 14417163975, 34256236039, 81499535281, 194130771581

Offset: 0

Author

N. J. A. Sloane, Nov 06 2003

Keywords

• nonn

Comments

The coat-hangers hang on a single rod and each coat-hanger may have 0, 1 or 2 coat-hangers hanging from it. There are n coat-hangers.

Arises when studying number of different configurations possible in a multiple star system.

Examples

			The eight possibilities with 4 edges are:
.||||..|||..|.|..||..||...|....|...|.
.......|.../.\...|...||../.\...|...|.
.................|.......|..../.\..|.
...................................|.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2542
• Piet Hut, Home Page
• Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, Toric Richardson varieties of Catalan type and Wedderburn-Etherington numbers, arXiv:2105.12274 [math.AG], 2021.

Crossrefs

Cf. A001190, A003214. Row sums of A088326.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*b(d+1),
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 11 2017

Mathematica

b[n_] := b[n] = If[n<2, n, If[OddQ[n], 0, Function[t, t*(1-t)/2][b[n/2]]] + Sum[b[i]*b[n-i], {i, 1, n/2}]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Formula

G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + ... = G001190(x)/x - 1 and G001190 is the g.f. for the Wedderburn-Etherington numbers A001190. - N. J. A. Sloane.

G.f.: 1/Product_{k>0} (1-x^k)^A001190(k+1). - Vladeta Jovovic, May 29 2005