OFFSET
1,5
COMMENTS
"Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.
Callan points out that A002792 is an incorrect version of this sequence. - Joerg Arndt, Jul 01 2014
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..415
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], 2014.
FORMULA
a(n) = A060356(n)/n.
1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.
E.g.f.: A(x) = Sum_{n>=0} a(n+1)*x^n/n! satisfies A(x) = exp(x*A(x))/(1+x). - Olivier Gérard, Dec 31 2013 (edited by Gus Wiseman, Dec 31 2019)
E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - Ilya Gutkovskiy, Jul 01 2020
MATHEMATICA
f[n_] := Sum[(-1)^(n-k)*n!/k!*Binomial[n-1, k-1]*k^(k-1), {k, n}]/n; Table[ f[n], {n, 20}] (* Robert G. Wilson v, Jul 21 2005 *)
PROG
(PARI) a(n) = { 1/n * sum(k=1, n, (-1)^(n-k) * binomial(n, k) * (n-1)!/(k-1)! * k^(k-1) ); } \\ Joerg Arndt, Aug 28 2014
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Jul 20 2005
EXTENSIONS
More terms from Robert G. Wilson v, Jul 21 2005
New name (from A002792) by Joerg Arndt, Aug 28 2014
Offset corrected by Gus Wiseman, Dec 31 2019
STATUS
approved