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A001059 Number of doubly labeled heap-ordered trees.

%I A001059 #30 Sep 17 2022 03:12:12
%S A001059 1,1,5,59,1263,42713,2094399,140434335,12340275539,1375857855221,
%T A001059 189751578038547,31714568837559539,6316261763436325285,
%U A001059 1477890415844440910325,401400487846091289175217,125247016772173387008904623,44493481073675052201518261955
%N A001059 Number of doubly labeled heap-ordered trees.
%C A001059 A standard heap-ordered tree with n+1 nodes is a finite rooted tree in which all the nodes except the root are labeled with the natural numbers between 1 and n, which satisfies the property that the labels of the children of a node are all larger than the label of the node.
%C A001059 Rooted trees counted by a(n) have n non-root vertices with two independent heap ordering labelings. The number of standard heap-ordered trees with n non-root vertices is A001147(n). - _Andrey Zabolotskiy_, Sep 16 2022
%H A001059 T. D. Noe, <a href="/A001059/b001059.txt">Table of n, a(n) for n = 0..100</a>
%H A001059 R. L. Grossman and R. G. Larson, <a href="https://arxiv.org/abs/0706.1327">Hopf Algebras of Heap Ordered Trees and Permutations</a>, arXiv:0706.1327 [math.RA], 2007.
%H A001059 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A001059 Doubly exponential generating function f(z) = Sum_{n>=0} a(n+1) z^n/n!^2 satisfies zf"+f'=1/(1-f). [Clarified by _Andrey Zabolotskiy_, Sep 16 2022]
%F A001059 a(n) = Sum_{k=0..n-1} binomial(n, k)^2*a(k)*a(n-k-1). - _Vladeta Jovovic_, Oct 22 2005
%t A001059 t = {1}; Do[AppendTo[t, Sum[Binomial[n, k]^2 t[[k+1]] t[[n-k]], {k, 0, n-1}]], {n, 20}] (* _T. D. Noe_, Jun 25 2012 *)
%Y A001059 Cf. A001147.
%K A001059 nonn
%O A001059 0,3
%A A001059 _Helmut Prodinger_
%E A001059 Name edited by _Andrey Zabolotskiy_, Sep 16 2022