A356259 - OEIS

A356259

Number of labeled rooted trees on [n] that have a primary branch.

0, 2, 6, 60, 500, 7290, 100842, 1978368, 38263752, 949218750, 23579476910, 709026379776, 21505924728444, 760772509715764, 27246730957031250, 1109165339867873280, 45798768824157052688, 2109518949433090534902

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

OFFSET

1,2

COMMENTS

This sequence is the labeled version of A027415 where the definition can be found.

LINKS

Table of n, a(n) for n=1..18.

N. J. A. Sloane, Illustration of initial terms

FORMULA

a(n) = Sum_{i=1..floor(n/2)} binomial(n,n-i)*r(n-i)*r(i) where r(i) = A000169(i).

a(n) = A000169(n) - A356073(n).

EXAMPLE

a(5) = 500. In the illustrations by Sloane found in the link above, for n = 5, there are A027415(5) = 6 rooted trees with a primary branch: the first six trees shown.

MATHEMATICA

Table[Sum[Binomial[n, n - i] (n - i)^(n - i - 1)*i^(i - 1), {i, 1, Floor[n/2]}], {n, 1, 20}]

CROSSREFS

Cf. A027415, A356074, A000169.

Sequence in context: A056604 A376515 A362702 * A215720 A376492 A371018

Adjacent sequences: A356256 A356257 A356258 * A356260 A356261 A356262

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Jul 31 2022

STATUS

approved