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A124973 a(n) = Sum_{k=0..(n-2)/2} a(k)a*(n-1-k), with a(0) = a(1) = 1.

%I A124973 #12 Feb 22 2021 03:36:12
%S A124973 1,1,1,1,2,3,6,11,22,42,87,174,365,745,1587,3303,7103,14974,32477,
%T A124973 69284,151172,325077,713400,1545719,3406989,7423648,16429555,35992438,
%U A124973 79912474,175785514,391488688,864591621,1930333822,4276537000
%N A124973 a(n) = Sum_{k=0..(n-2)/2} a(k)a*(n-1-k), with a(0) = a(1) = 1.
%C A124973 Number of unordered rooted trees with all outdegrees <= 2 and, if a node has two subtrees, they have a different number of nodes (equivalently, ordered rooted trees where the left subtree has more nodes than the right subtree).
%H A124973 G. C. Greubel, <a href="/A124973/b124973.txt">Table of n, a(n) for n = 0..1000</a>
%F A124973 Lim_{n->infinity} a(n)^(1/n) = 2.327833478... - _Vaclav Kotesovec_, Nov 20 2019
%p A124973 a:= proc(n) option remember;
%p A124973       if n<2 then 1
%p A124973     else add(a(j)*a(n-j-1), j=0..floor((n-2)/2))
%p A124973       fi
%p A124973     end:
%p A124973 seq(a(n), n=0..40); # _G. C. Greubel_, Nov 19 2019
%t A124973 a[n_]:= a[n]= If[n<2, 1, Sum[a[j]*a[n-j-1], {j, 0, (n-2)/2}]]; Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Nov 19 2019 *)
%o A124973 (PARI) a(n) = if(n<2, 1, sum(j=0, (n-2)\2, a(j)*a(n-j-1))); \\ _G. C. Greubel_, Nov 19 2019
%o A124973 (Sage)
%o A124973 @CachedFunction
%o A124973 def a(n):
%o A124973     if (n<2): return 1
%o A124973     else: return sum(a(j)*a(n-j-1) for j in (0..floor((n-2)/2)))
%o A124973 [a(n) for n in (0..40)] # _G. C. Greubel_, Nov 19 2019
%Y A124973 Cf. A000108, A000992, A001190, A032305.
%K A124973 easy,nonn
%O A124973 0,5
%A A124973 _Franklin T. Adams-Watters_, Nov 14 2006