A095338 -id:A095338 - cp's OEIS frontend

A275462 Number of leaves in all simple labeled connected graphs on n nodes.

0, 0, 2, 6, 48, 760, 21840, 1121568, 104510336, 18111498624, 5966666196480, 3794613745429760, 4704698796461841408, 11443317008255593064448, 54831540882238864189229056, 519046250316393184411087165440, 9726643425055315256306341282775040

Offset: 0

Geoffrey Critzer, Jul 28 2016

nonn

A leaf is a vertex of degree 1.

Cf. A095338.

b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
a:= n-> n*(n-1)*b(n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 31 2016

nn = 15; Clear[f];f[z_] := Sum[2^Binomial[n, 2] z^n/n!, {n, 0, nn + 1}];Range[0, nn]! CoefficientList[Series[ z z D[Log[f[z]], z] , {z, 0, nn}], z]

E.g.f.: x*A(x) = x^2* d[log(B(x))]/dx where A(x) is the e.g.f. for A053549 and B(x) is the e.g.f. for A006125.

For n>=1, a(n) = n*(n-1)*A001187(n-1).

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

Geoffrey Critzer, Apr 20 2017

			1,
2,   2,
6,   12,   6,
32,  96,   96,   32,
320, 1280, 1920, 1280, 320,
...

Row sums give A095340.

Columns for k=0-3: A123903, A095338, A038094, A038096.

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

E.g.f. for column k: x * Sum_{n>=0} binomial(n,k)*2^binomial(n,2)*x^n/n!.
Sum_{k=1..n-1} T(n,k)*k/2 = A095351(n).
T(n,k) = n*binomial(n-1,k)*2^binomial(n-1,2). - Alois P. Heinz, Apr 21 2017

cp's OEIS Frontend

A275462 Number of leaves in all simple labeled connected graphs on n nodes.

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