OFFSET
1,2
COMMENTS
A rooted trimmed tree is a tree with a forbidden limb of length 2.
A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.
LINKS
FORMULA
E.g.f. satisfies A(x) = x*exp(A(x) - x^2).
E.g.f.: -LambertW(-x/exp(x^2)). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Jan 08 2014
MAPLE
A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x, n), polynom) fi end: a:= n-> coeff(A(n+1), x, n)*n!: seq(a(n), n=1..25); # Alois P. Heinz, Aug 23 2008
MATHEMATICA
a[n_] := Sum[ Boole[ EvenQ[n-m]]*(m^((n+m)/2-2)/((n-m)/2)!)*((-1)^((n-m)/2)/(m-1)!), {m, 1, n}]*n!; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Sep 10 2012, after Vladimir Kruchinin *)
Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)], {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
PROG
(Maxima) a(n):=sum((if mod(n-m, 2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!, m, 1, n); /* Vladimir Kruchinin, Aug 07 2012 */
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
Christian G. Bower, Dec 15 1999
STATUS
approved