A088326 -id:A088326

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A036658

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

+0
3

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

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

OFFSET

0,7

LINKS

Table of n, a(n) for n=0..35.

Index entries for sequences related to rooted trees

FORMULA

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)).

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A088326, A036656.

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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.

+0
4

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

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

Cf. A001190, A003214. Row sums of A088326.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 06 2003

STATUS

approved

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.

+0
2

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

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

OFFSET

0,3

LINKS

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

FORMULA

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

CROSSREFS

Cf. A088326.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 11 2018

STATUS

approved

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