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
A038095
Number of rooted connected graphs on n labeled nodes where the root has degree 2.
Original entry on oeis.org
6, 72, 1440, 49680, 2998800, 324237312, 64440883584, 24059497893120, 17143668999705600, 23569875858252303360, 63000019679242001900544, 329150325651711743768150016, 3374625529825460904919664793600, 68094821953233373962606732799672320
Offset: 3
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seq(n)={Vec(serlaplace(sum(k=1, n, k*binomial(k-1, 2)*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
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
A285529
Triangle read by rows: T(n,k) is the number of nodes of degree k counted over all simple labeled graphs on n nodes, n>=1, 0<=k<=n-1.
Original entry on oeis.org
1, 2, 2, 6, 12, 6, 32, 96, 96, 32, 320, 1280, 1920, 1280, 320, 6144, 30720, 61440, 61440, 30720, 6144, 229376, 1376256, 3440640, 4587520, 3440640, 1376256, 229376, 16777216, 117440512, 352321536, 587202560, 587202560, 352321536, 117440512, 16777216
Offset: 1
1,
2, 2,
6, 12, 6,
32, 96, 96, 32,
320, 1280, 1920, 1280, 320,
...
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nn = 9; Map[Select[#, # > 0 &] &,
Drop[Transpose[Table[A[z_] := Sum[Binomial[n, k] 2^Binomial[n, 2] z^n/n!, {n, 0, nn}];Range[0, nn]! CoefficientList[Series[z A[z], {z, 0, nn}], z], {k,0, nn - 1}]], 1]] // Grid
Showing 1-4 of 4 results.