OFFSET
0,3
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.
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.
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
EXAMPLE
The eight possibilities with 4 edges are:
.||||..|||..|.|..||..||...|....|...|.
.......|.../.\...|...||../.\...|...|.
.................|.......|..../.\..|.
...................................|.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 06 2003
STATUS
approved