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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.

%I A088325 #17 Sep 08 2021 21:51:34
%S A088325 1,1,2,4,8,16,34,71,153,332,730,1617,3620,8148,18473,42097,96420,
%T A088325 221770,512133,1186712,2758707,6431395,15033320,35224825,82720273,
%U A088325 194655030,458931973,1083926784,2564305754,6075896220,14417163975,34256236039,81499535281,194130771581
%N 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.
%C A088325 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.
%C A088325 Arises when studying number of different configurations possible in a multiple star system.
%H A088325 Alois P. Heinz, <a href="/A088325/b088325.txt">Table of n, a(n) for n = 0..2542</a>
%H A088325 Piet Hut, <a href="http://www.sns.ias.edu/~piet/">Home Page</a>
%H A088325 Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, <a href="https://arxiv.org/abs/2105.12274">Toric Richardson varieties of Catalan type and Wedderburn-Etherington numbers</a>, arXiv:2105.12274 [math.AG], 2021.
%F A088325 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_.
%F A088325 G.f.: 1/Product_{k>0} (1-x^k)^A001190(k+1). - _Vladeta Jovovic_, May 29 2005
%e A088325 The eight possibilities with 4 edges are:
%e A088325 .||||..|||..|.|..||..||...|....|...|.
%e A088325 .......|.../.\...|...||../.\...|...|.
%e A088325 .................|.......|..../.\..|.
%e A088325 ...................................|.
%p A088325 b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
%p A088325       (t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
%p A088325     end:
%p A088325 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*b(d+1),
%p A088325       d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A088325     end:
%p A088325 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 11 2017
%t A088325 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}]];
%t A088325 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
%t A088325 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 11 2018, after _Alois P. Heinz_ *)
%Y A088325 Cf. A001190, A003214. Row sums of A088326.
%K A088325 nonn
%O A088325 0,3
%A A088325 _N. J. A. Sloane_, Nov 06 2003