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 A108919 Number of series-reduced labeled trees with n nodes. 11
 1, 0, 1, 1, 13, 51, 601, 4803, 63673, 775351, 12186061, 196158183, 3661759333, 72413918019, 1583407093633, 36916485570331, 929770285841137, 24904721121298671, 711342228666833173, 21502519995056598639, 687345492498807434461, 23135454269839313430715, 818568166383797223246601, 30357965273255025673685091 (list; graph; refs; listen; history; text; internal format)
 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 The rooted version is A060356. Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410. Sequence in context: A022283 A135971 A146143 * A002792 A183941 A022673 Adjacent sequences: A108916 A108917 A108918 * A108920 A108921 A108922 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

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Last modified September 25 05:35 EDT 2023. Contains 365582 sequences. (Running on oeis4.)