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%I #19 Dec 19 2015 12:42:21
%S 0,1,1,2,5,19,90,520,3475,26550,228050,2177020,22860090,261870070,
%T 3249793360,43432062300,621911561150,9498946124800,154152712434600,
%U 2648808048264400,48043086765929200,917249983543337400
%N Number of increasing rooted trees with a forbidden limb of length 3.
%C In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
%C 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.
%H Vaclav Kotesovec, <a href="/A052324/b052324.txt">Table of n, a(n) for n = 0..240</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F E.g.f. satisfies A'(x) = exp(A(x) - x^3/6). - corrected by _Vaclav Kotesovec_, Mar 28 2014
%F a(n) ~ d^n * (n-1)!, where d = 0.9546118344740519430556804... - _Vaclav Kotesovec_, Mar 28 2014
%F In closed form, d = 1/r, where r = 1.04754620033697244977759528695194261... is the root of the equation 1 = Integral_{x=0..r} exp(-x^3/6) dx. - _Vaclav Kotesovec_, Aug 21 2014
%t CoefficientList[Assuming[{Element[x, Reals], x > 0}, Series[-Log[1-6^(1/3)*Gamma[1/3]/3 + 1/3*x*ExpIntegralE[2/3, x^3/6]], {x, 0, 20}]], x]*Range[0, 20]! (* _Vaclav Kotesovec_, Mar 28 2014 *)
%o (PARI) {a(n)=local(A=x); for(i=0, n, A=intformal(exp(A-x^3/6+O(x^n)) )); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Mar 28 2014
%Y Cf. A002955, A002988-A002992, A052319-A052329.
%K nonn,eigen
%O 0,4
%A _Christian G. Bower_, Dec 15 1999
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