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

Author

Geoffrey Critzer, Jul 28 2016

Keywords

• nonn

Comments

A leaf is a vertex of degree 1.

Crossrefs

Cf. A095338.

Programs

Maple

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

Mathematica

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]

Formula

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).