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A330951 Number of singleton-reduced unlabeled rooted trees with n nodes. 9
1, 1, 1, 3, 5, 11, 24, 52, 119, 272, 635, 1499, 3577, 8614, 20903, 51076, 125565, 310302, 770536, 1921440, 4809851, 12081986, 30445041, 76938794, 194950040, 495174037, 1260576786, 3215772264, 8219437433, 21046602265, 53982543827, 138678541693, 356785641107 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: A(x) satisfies A(x) = x + x*exp(Sum_{k>=1} A(x^k)/k) - x*exp(Sum_{k>=1} x^k*A(x^k)/(1 + x^k)/k). - Andrew Howroyd, Dec 10 2020

a(n) ~ c * d^n / n^(3/2), where d = 2.69474016697407303512228736537683134987637576... and c = 0.41800971384719166056172258174139385922545... - Vaclav Kotesovec, Nov 16 2021

EXAMPLE

The a(1) = 1 through a(6) = 11 trees:

  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)

                ((oo))  ((ooo))   ((oooo))

                (o(o))  (o(oo))   (o(ooo))

                        (oo(o))   (oo(oo))

                        ((o(o)))  (ooo(o))

                                  ((o)(oo))

                                  ((o(oo)))

                                  ((oo(o)))

                                  (o((oo)))

                                  (o(o)(o))

                                  (o(o(o)))

MATHEMATICA

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]], {ptn, IntegerPartitions[n-1]}];

Table[Length[Select[urt[n], FreeQ[#, q:{__List}/; Times@@Length/@q==1]&]], {n, 10}]

PROG

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

seq(n)={my(v=vector(n)); v[1]=1; for(n=1, #v-1, v[n+1] = EulerT(v[1..n])[n] - EulerT(Vec(x^2*Ser(v[1..n-1])/(1+x), -n))[n]); v} \\ Andrew Howroyd, Dec 10 2020

CROSSREFS

The Matula-Goebel numbers of these trees are given by A330943.

The series-reduced case is A001678.

Unlabeled rooted trees are counted by A000081.

Singleton-reduced phylogenetic trees are A000311.

Cf. A000669, A003238, A004111, A324694.

Sequence in context: A074874 A051439 A214873 * A018116 A259643 A167796

Adjacent sequences:  A330948 A330949 A330950 * A330952 A330953 A330954

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2020

EXTENSIONS

Terms a(19) and beyond from Andrew Howroyd, Dec 10 2020

STATUS

approved

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Last modified December 5 03:59 EST 2021. Contains 349530 sequences. (Running on oeis4.)