OFFSET
1,4
COMMENTS
In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
A trimmed tree is a tree with a forbidden limb of length 2.
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.
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
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..465
M. E. Jones and J. B. Remmel, Pattern matching in the cycle structures of permutations, Pure Math. Appl. (PU.M.A.), 22 (2011) 173-208.
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021, p. 7.
FORMULA
E.g.f.: A(x) = 1/B(-x) where B'(x) is e.g.f. of A006882 and B(0) = 1.
E.g.f.: A(x) satisfies A'(x) = exp(A(x)-x^2/2).
E.g.f.: exp(-x^2/2)/(1-int[0..x, exp(-x^2/2)]). - Ralf Stephan, Apr 25 2004
E.g.f.: -log(1-sqrt(Pi/2)*erf(x/sqrt(2))). - Vaclav Kotesovec, Jan 07 2014
Limit n->infinity (a(n)/n!)^(1/n) = 1/(sqrt(2)*InverseErf(sqrt(2/Pi))) = 1/A240885 = 0.7839769312035474991... - Vaclav Kotesovec, Jan 07 2014
a(n) ~ (n-1)! / (sqrt(2)*InverseErf(sqrt(2/Pi)))^n. - Vaclav Kotesovec, Aug 22 2014
MAPLE
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
MATHEMATICA
Rest[CoefficientList[Series[-Log[1-Sqrt[Pi/2]*Erf[x/Sqrt[2]]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 07 2014 *)
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
Christian G. Bower, Dec 11 1999
EXTENSIONS
Formula updated by Christian G. Bower, Mar 06 2001
STATUS
approved