%I #15 Nov 23 2020 20:29:04
%S 32,1120,53760,4155200,550305280,129990260736,56369709634560,
%T 45808126727193600,70779622448719134720,210103333009795315650560,
%U 1207180278201294640467288064,13500153139563947729371140096000,295095590701444457972767937903329280
%N Number of rooted connected graphs on n labeled nodes where the root has degree 3.
%H Andrew Howroyd, <a href="/A038097/b038097.txt">Table of n, a(n) for n = 4..50</a>
%F E.g.f.: B(x)/C(x) where B, C respectively are the e.g.f.'s for A038096 and A006125.
%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.
%o (PARI) seq(n)={Vec(serlaplace(sum(k=1, n, k*binomial(k-1,3)*2^binomial(k-1,2)*x^k/k!)/sum(k=0, n, 2^binomial(k, 2)*x^k/k!) + O(x*x^n)))} \\ _Andrew Howroyd_, Nov 23 2020
%Y Cf. A038094, A038095, A038096.
%K nonn,easy
%O 4,1
%A _Christian G. Bower_, Jan 04 1999; suggested by Vlady Ravelomanana
%E Terms a(13) and beyond corrected by _Andrew Howroyd_, Nov 23 2020
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