A355023 Number of labeled trees on n nodes with maximum degree three and three vertices of degree three.

5040, 317520, 12700800, 419126400, 12573792000, 359610451200, 10069092633600, 280496151936000, 7853892254208000, 222526947202560000, 6408776079433728000, 188184970332463104000, 5645549109973893120000, 173274930375352565760000, 5445783526082509209600000, 175354229539856796549120000

Offset

8,1

Links

Table of n, a(n) for n=8..23.

Marko R. Riedel et al., Trees with maximum degree three and three vertices of degree three, Mathematics Stack Exchange, 2022.

Formula

a(n) = (1/8)*n!*binomial(n-2,n-8).

E.g.f.: x^8/(8*(1 - x)^7). - Stefano Spezia, Jun 16 2022

a(n) = 7*binomial(n,n-8)*(n-2)!. - Chai Wah Wu, Jun 16 2022

From Amiram Eldar, Oct 23 2025: (Start)

Sum_{n>=8} 1/a(n) = 154267/11025 - 524*e/105 + 34*(gamma - Ei(1))/105, where e = A001113, Ei(1) = A091725, and gamma = A001620.

Sum_{n>=8} (-1)^n/a(n) = 3092*(gamma - Ei(-1))/35 - 368/(7*e) - 187549/3675, where Ei(-1) = -A099285. (End)

Example

First term counts (the nodes are labeled for a total of 8! possibilities divided by eight automorphisms, 5040):
          o
          |
          |
          |
          |
          o
         / \
        /   \
       o     o
      / \   / \
     o   o o   o

Mathematica

CoefficientList[Series[x^8/(8(1-x)^7), {x, 0, 23}], x]
Table[n!, {n, 0, 23}] (* Stefano Spezia, Jun 16 2022 *)

Prog

(Python)
from math import comb, factorial
def A355023(n): return 7*comb(n, n-8)*factorial(n-2) # Chai Wah Wu, Jun 16 2022
(PARI) a(n) = 7*binomial(n, n-8)*(n-2)! \\ Felix Fröhlich, Jun 17 2022

Crossrefs

Cf. A355024.

Cf. A001620, A001113, A091725, A099285.

Sequence in context: A254080 A228910 A258419 * A179062 A342075 A055362

Adjacent sequences: A355020 A355021 A355022 * A355024 A355025 A355026

Keyword

nonn,easy

Author

Marko Riedel, Jun 15 2022

Status

approved