A101313 Number of painted forests - exactly one of its trees is painted - on labeled vertex set [n].

1, 3, 12, 68, 525, 5262, 65674, 987408, 17426565, 353759300, 8127640224, 208600774032, 5917247520457, 183872561612040, 6212370268252950, 226762373954676608, 8893485959056048521, 372980176625914811568, 16656844860594186642100, 789196576594282265505600

Offset: 1

Author

Joseph G. Moser (jmoser(AT)wcupa.edu), Jan 26 2005

Keywords

• nonn

Examples

			a(5) = 291 + (16*4*1)+(3*6*3)+(1*4*12)+(1*1*68) = 525.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100

Programs

Maple

B:= n-> exp(add(k^(k-2) *x^k/k!, k = 1..n )): b:= n-> coeff(series(B(n), x,n+1), x,n)*n!: a:= n-> add(binomial(n,k) *k^(k-2) *b(n-k), k=1..n): seq(a(n), n=1..25);  # Alois P. Heinz, Sep 10 2008

Mathematica

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Drop[Range[0,nn]!CoefficientList[Series[D[Exp[y(t-t^2/2)],y]/.y->1,{x,0,nn}],x],1]  (* Geoffrey Critzer, Nov 04 2012 *)

Formula

a(n) = f(n) + SUM{((n-i)^(n-i-2))*C((n-1), i)*a(i):i=1, 2, ..(n-1)}, where f(n)=number of forests on labeled vertex set [n], A001858.

Exponential convolution of A000272 and A001858: a(n) = Sum_{k=1..n} binomial(n, k)*k^(k-2)*A001858(n-k). E.g.f.: B(x)*exp(B(x)), where B(x) is e.g.f. for A000272. - Vladeta Jovovic, May 24 2005

a(n) = Sum_{m=1..n} A105599(n,m)*m. - Geoffrey Critzer, Nov 04 2012

Extensions

More terms from Alois P. Heinz, Sep 10 2008