login
Number of painted forests - exactly one of its trees is painted - on labeled vertex set [n].
4

%I #18 Dec 29 2014 18:37:15

%S 1,3,12,68,525,5262,65674,987408,17426565,353759300,8127640224,

%T 208600774032,5917247520457,183872561612040,6212370268252950,

%U 226762373954676608,8893485959056048521,372980176625914811568,16656844860594186642100,789196576594282265505600

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

%H Alois P. Heinz, <a href="/A101313/b101313.txt">Table of n, a(n) for n = 1..100</a>

%F 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.

%F 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

%F a(n) = Sum_{m=1..n} A105599(n,m)*m. - _Geoffrey Critzer_, Nov 04 2012

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

%p 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

%t 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 *)

%K nonn

%O 1,2

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

%E More terms from _Alois P. Heinz_, Sep 10 2008