A088326 -id:A088326 - cp's OEIS frontend

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

N. J. A. Sloane, Nov 06 2003

nonn

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.

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

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.

Cf. A001190, A003214. Row sums of A088326.

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

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 *)

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

A036658 Number of n-node rooted unlabeled trees with exactly 3 edges at root and otherwise out-degree <= 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 6, 14, 29, 68, 147, 337, 757, 1734, 3953, 9113, 20988, 48645, 112909, 263084, 614201, 1438001, 3373253, 7930660, 18679005, 44075988, 104173194, 246604137, 584620470, 1387879434, 3299067379, 7851736348, 18708682855, 44627133541, 106563177864

Offset: 0

N. J. A. Sloane

Index entries for sequences related to rooted trees

Cf. A088326, A036656.

CI2 := proc(f) (1/2)*(f^2+subs(x=x^2,f)); end; CI3 := proc(f) (1/6)*(f^3+3*subs(x=x^2,f)*f+2*subs(x=x^3,f)); end;
N := 40: G036658 := series(x^3*CI3(G036656),x,N); A036658 := n->coeff(G036658,x,n);

terms = 35;
CI3[f_] := (1/3!)*(f^3 + 3*(f /. x -> x^2)*f + 2*(f /. x -> x^3));
G036656[] = 0; Do[G036656[x] = x + (1/2)*(G036656[x]^2 + G036656[x^2]) + O[x]^terms // Normal, terms];
G036658[x_] = x^3*CI3[G036656[x] - x] + O[x]^(terms+5);
Drop[CoefficientList[G036658[x], x], 5] (* Jean-François Alcover, Jan 24 2018, adapted from Maple *)

Let G036656(x) = g.f. for A036656. G.f.: x^3*cycle_index(S3, G036656), where cycle_index(Sk, f) means apply the cycle index for the symmetric group S_k to f(x).

E.g., cycle_index(S2, f) = (1/2!)*(f^2+subs(x=x^2, f), cycle_index(S3, f) = (1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)).

Corrected by N. J. A. Sloane, May 03 2000

A305839 Number of unlabeled forests of rooted trees with 2n edges and n connected components, in which the outdegree of each node is <= 2 and the symmetric group acts on the components.

Original entry on oeis.org

1, 1, 3, 6, 15, 32, 77, 172, 405, 930, 2180, 5070, 11914, 27929, 65829, 155202, 367053, 868990, 2061723, 4897502, 11652547, 27757960, 66210042, 158103242, 377957299, 904439542, 2166408422, 5193894809, 12463003846, 29929966312, 71933014935, 173009938416

Offset: 0

Alois P. Heinz, Jun 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A088326.

a(n) = A088326(2n,n) = A088326(2n+k,n+k) for k >= 0.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A036658 Number of n-node rooted unlabeled trees with exactly 3 edges at root and otherwise out-degree <= 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A305839 Number of unlabeled forests of rooted trees with 2n edges and n connected components, in which the outdegree of each node is <= 2 and the symmetric group acts on the components.

Views

Author

Keywords

Links

Crossrefs

Formula