%I #37 Aug 06 2021 02:33:12
%S 1,1,1,2,7,28,131,720,4513,31824,249513,2151744,20242983,206313024,
%T 2264425179,26628836352,334022337153,4451717814528,62820790592913,
%U 935750983412736,14672143677452679,241555066200437760
%N Number of increasing rooted trimmed trees with n nodes.
%C In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
%C A trimmed tree is a tree with a forbidden limb of length 2.
%C A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.
%C Number of permutations on [n+1] beginning with 12 and avoiding a consecutive 132 pattern (n>=1). For example, a(4)=2 counts 12345, 12453. - _Ralf Stephan_, Apr 25 2004
%H Vaclav Kotesovec, <a href="/A052319/b052319.txt">Table of n, a(n) for n = 1..465</a>
%H M. E. Jones and J. B. Remmel, <a href="http://puma.dimai.unifi.it/22_2/jones_remmel.pdf">Pattern matching in the cycle structures of permutations</a>, Pure Math. Appl. (PU.M.A.), 22 (2011) 173-208.
%H S. Kitaev, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s48kitaev.html">Generalized pattern avoidance with additional restrictions</a>, Sem. Lothar. Combinat. B48e (2003).
%H S. Kitaev and T. Mansour, <a href="https://arxiv.org/abs/math/0205182">Simultaneous avoidance of generalized patterns</a>, arXiv:math/0205182 [math.CO], 2002.
%H Rupert Li, <a href="https://arxiv.org/abs/2107.12353">Vincular Pattern Avoidance on Cyclic Permutations</a>, arXiv:2107.12353 [math.CO], 2021, p. 7.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F E.g.f.: A(x) = 1/B(-x) where B'(x) is e.g.f. of A006882 and B(0) = 1.
%F E.g.f.: A(x) satisfies A'(x) = exp(A(x)-x^2/2).
%F E.g.f.: exp(-x^2/2)/(1-int[0..x, exp(-x^2/2)]). - _Ralf Stephan_, Apr 25 2004
%F E.g.f.: -log(1-sqrt(Pi/2)*erf(x/sqrt(2))). - _Vaclav Kotesovec_, Jan 07 2014
%F Limit n->infinity (a(n)/n!)^(1/n) = 1/(sqrt(2)*InverseErf(sqrt(2/Pi))) = 1/A240885 = 0.7839769312035474991... - _Vaclav Kotesovec_, Jan 07 2014
%F a(n) ~ (n-1)! / (sqrt(2)*InverseErf(sqrt(2/Pi)))^n. - _Vaclav Kotesovec_, Aug 22 2014
%p seq(n! * coeff(series(-log(1-sqrt(Pi/2)*erf(x/sqrt(2))), x, n+1), x, n), n=1..20) # _Vaclav Kotesovec_, Jan 07 2014
%t Rest[CoefficientList[Series[-Log[1-Sqrt[Pi/2]*Erf[x/Sqrt[2]]], {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 07 2014 *)
%Y Cf. A002955, A002988-A002992, A052318-A052329.
%K nonn,eigen
%O 1,4
%A _Christian G. Bower_, Dec 11 1999
%E Formula updated by _Christian G. Bower_, Mar 06 2001