A038096 - OEIS

%I #22 Nov 23 2020 21:02:15

%S 32,1280,61440,4587520,587202560,135291469824,57724360458240,

%T 46443371157258240,71337018097548656640,211030752203237270487040,

%U 1210134745434243803880882176,13518305228996352601898436526080

%N Number of rooted graphs on n labeled nodes where the root has degree 3.

%C The graphs are not necessarily connected. The nodes are labeled.

%H Andrew Howroyd, <a href="/A038096/b038096.txt">Table of n, a(n) for n = 4..50</a>

%F a(n) = n*binomial(n-1,3)*2^binomial(n-1,2). (There are n choices for the root, binomial(n-1,3) choices for the nodes it joined to, and 2^binomial(n-1,2) choices for the edges between the non-root nodes.)

%e For n=4, take 4 nodes labeled a,b,c,d. We can choose the root in 4 ways, say a, and it must be joined to b,c,d. Each of the three edges bc, bd, cd may or may not exist, so there are 4*8 = 32 = a(4) possibilities.

%t Table[n Binomial[n-1,3] 2^Binomial[n-1,2],{n,4,20}] (* _Harvey P. Dale_, Sep 14 2011 *)

%o (PARI) a(n) = {n*binomial(n-1,3)*2^binomial(n-1,2)} \\ _Andrew Howroyd_, Nov 23 2020

%Y Cf. A006125, A038094, A038095, A038097.

%K nonn,easy

%O 4,1

%A _Christian G. Bower_, Jan 04 1999

%E Edited by _N. J. A. Sloane_, Sep 14 2011