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%I #15 Apr 19 2026 14:54:27
%S 1,1,2,2,2,4,8,4,10,28,36,64,112,200,496,888,1778,3644,9644,18384,
%T 35244,88512,240912,490496,1068256,2898384,8330688,18072064,42690600,
%U 119723792,362082648,837947696,2118951762,6211260556,19630042716,48172034512,128288363220,392572281584
%N Number of alpha-labelings of the path tree on n vertices.
%C A labeling of a tree on n vertices is called an alpha-labeling if it is a graceful labeling with the additional property that there exists a critical value k such that for every edge u--v, one endpoint has label <= k and the other has label > k.
%C A path tree is a graph with two vertices of degree 1 and all other vertices of degree 2.
%H Bert Dobbelaere, <a href="/A395104/b395104.txt">Table of n, a(n) for n = 1..55</a>
%H Igor Blokhin, <a href="https://github.com/IgorBlokhin/Graph-Theory">Graph Theory</a> (Python repository).
%H A. Rosa, <a href="https://www.researchgate.net/publication/244474213_On_certain_valuations_of_the_vertices_of_a_graph">On certain valuations of the vertices of a graph</a>, Theory of Graphs (International Symposium, Rome, 1966), Gordon and Breach, 1967, pp. 349-355.
%H David A. Sheppard, <a href="https://doi.org/10.1016/0012-365X(76)90051-0">The factorial representation of balanced labelled graphs</a>, Discrete Math., 15 (1976), no. 4, 379-388.
%e For example, consider the path graph on 4 vertices. The labeling 0--3--1--2 is graceful since the induced edge labels are |0-3|=3, |3-1|=2, |1-2|=1, giving {1,2,3}. Moreover, it is an alpha-labeling: taking k=1, the vertices are split into {0,1} and {2,3}, and every edge has one endpoint in each part.
%Y Cf. A112362, A005193, A392797.
%K nonn,hard
%O 1,3
%A _Igor Blokhin_, Apr 11 2026
%E More terms from _Bert Dobbelaere_, Apr 19 2026