A355023 - OEIS

%I #34 Aug 22 2022 04:49:35

%S 5040,317520,12700800,419126400,12573792000,359610451200,

%T 10069092633600,280496151936000,7853892254208000,222526947202560000,

%U 6408776079433728000,188184970332463104000,5645549109973893120000,173274930375352565760000,5445783526082509209600000,175354229539856796549120000

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

%H Marko R. Riedel et al., Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4472439/">Trees with maximum degree three and three vertices of degree three</a>.

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

%F E.g.f.: x^8/(8*(1 - x)^7). - _Stefano Spezia_, Jun 16 2022

%F a(n) = 7*binomial(n,n-8)*(n-2)!. - _Chai Wah Wu_, Jun 16 2022

%e First term counts (the nodes are labeled for a total of 8! possibilities divided by eight automorphisms, 5040):

%e           o

%e           |

%e           |

%e           |

%e           |

%e           o

%e          / \

%e         /   \

%e        o     o

%e       / \   / \

%e      o   o o   o

%t CoefficientList[Series[x^8/(8(1-x)^7), {x,0,23}], x]

%t Table[n!, {n, 0, 23}] (* _Stefano Spezia_, Jun 16 2022 *)

%o (Python)

%o from math import comb, factorial

%o def A355023(n): return 7*comb(n,n-8)*factorial(n-2) # _Chai Wah Wu_, Jun 16 2022

%o (PARI) a(n) = 7*binomial(n,n-8)*(n-2)! \\ _Felix Fröhlich_, Jun 17 2022

%Y Cf. A355024.

%K nonn

%O 8,1

%A _Marko Riedel_, Jun 15 2022