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%I #16 Dec 23 2020 19:49:16
%S 1,1,2,6,24,108,596,3674,26068,202470,1753884,16435754,168174596,
%T 1842418704,21757407158,272771715272,3649515044178,51532670206504,
%U 770442883634326,12093451621846094,199856952123506304,3452120352032161404,62471981051497913826,1177664861561125869100,23163177237781937250558
%N Number of rooted graceful labelings with n edges.
%C A graceful labeling of a graph with n edges assigns distinct labels l(v) to the vertices such that 0<=l(v)<=n and the n differences |l(u)-l(v)| between labels of adjacent vertices are distinct. It is rooted if, in addition, either |l(u)-l(w)|>|l(u)-l(v)| for some neighbor of u or |l(v)-l(w)|>|l(u)-l(v)| for some neighbor of v, whenever |l(u)-l(v)|<n.
%C To generate such a labeling, choose one of the k+1 edges 0--(n-k), 1--(n+1-k), ..., k--(n-k), for each k=0, 1, ..., n-1, provided that at least one of the endpoints has already appeared in a previously chosen edge when k>0.
%H David A. Sheppard, <a href="http://dx.doi.org/10.1016/0012-365X(76)90051-0">The factorial representation of major balanced labelled graphs</a>, Discrete Math., Vol. 15, No. 4 (1976), 379-388.
%e a(5) = 108 < 120 = 5!, because 0--5,0--4,0--3,3--5,1--2 and 0--5,1--5,2--5,0--2,1--3 are forbidden, as well as five each beginning with 0--5,0--4,2--5,1--3 and 0--5,1--4,0--3,2--4.
%o (Python)
%o def solve(d, m_in):
%o ....global _n, _cache
%o ....args = (d, m_in)
%o ....if args in _cache:
%o ........return _cache[args]
%o ....if d == 0:
%o ........rv = 1
%o ....else:
%o ........rv = 0
%o ........m_test = 1 | (1<<d)
%o ........for p in range(_n + 1 - d):
%o ............if m_in & m_test:
%o ................rv += solve(d-1, m_in | m_test)
%o ............m_test <<= 1
%o ...._cache[args] = rv
%o ....return rv
%o def a338987(n):
%o ....global _cache, _n
%o ...._cache, _n = {}, n
%o ....return 1 if n<2 else 2*solve(n-2, 3|(1<<n))
%o # _Bert Dobbelaere_, Dec 23 2020
%Y Without rootedness, the number of graceful labelings is n!, A000142, as observed by Sheppard in 1976.
%K nonn
%O 0,3
%A _Don Knuth_, Dec 20 2020
%E a(17)-a(24) from _Bert Dobbelaere_, Dec 23 2020
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