A038094
Number of rooted graphs on n labeled nodes where the root has degree 2.
Original entry on oeis.org
6, 96, 1920, 61440, 3440640, 352321536, 67645734912, 24739011624960, 17416264183971840, 23779006032516218880, 63309225660971181146112, 330036748754793764694786048, 3379576307249088150474609131520
Offset: 3
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[n*Binomial(n-1, 2)*2^Binomial(n-1, 2): n in [3..20]]; // Vincenzo Librandi, Mar 29 2014
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Table[n*Binomial[n-1, 2]*2^Binomial[n-1, 2], {n, 3, 20}] (* Vaclav Kotesovec, Mar 29 2014 *)
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a(n) = {n * binomial(n-1, 2) * 2^binomial(n-1, 2)} \\ Andrew Howroyd, Nov 23 2020
A038096
Number of rooted graphs on n labeled nodes where the root has degree 3.
Original entry on oeis.org
32, 1280, 61440, 4587520, 587202560, 135291469824, 57724360458240, 46443371157258240, 71337018097548656640, 211030752203237270487040, 1210134745434243803880882176, 13518305228996352601898436526080
Offset: 4
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.
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Table[n Binomial[n-1,3] 2^Binomial[n-1,2],{n,4,20}] (* Harvey P. Dale, Sep 14 2011 *)
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a(n) = {n*binomial(n-1,3)*2^binomial(n-1,2)} \\ Andrew Howroyd, Nov 23 2020
A038097
Number of rooted connected graphs on n labeled nodes where the root has degree 3.
Original entry on oeis.org
32, 1120, 53760, 4155200, 550305280, 129990260736, 56369709634560, 45808126727193600, 70779622448719134720, 210103333009795315650560, 1207180278201294640467288064, 13500153139563947729371140096000, 295095590701444457972767937903329280
Offset: 4
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.
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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
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