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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois Alcover, Jun 11 2018, after Alois P. Heinz *)

CROSSREFS

Cf. A001190, A003214. Row sums of A088326.

Sequence in context: A273972 A275443 A288170 * A215930 A288260 A006210

Adjacent sequences:  A088322 A088323 A088324 * A088326 A088327 A088328

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 06 2003

STATUS

approved

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Last modified September 24 20:09 EDT 2018. Contains 315356 sequences. (Running on oeis4.)