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 A033472 Number of n-vertex labeled graphs that are gracefully labeled trees. 5
 1, 1, 2, 4, 12, 40, 164, 752, 4020, 23576, 155632, 1112032, 8733628, 73547332, 670789524, 6502948232, 67540932632, 740949762580, 8634364751264, 105722215202120, 1366258578159064, 18468456090865364, 262118487952306820 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A graph with n edges is graceful if its vertices can be labeled with distinct integers in the range 0,1,...,n in such a way that when the edges are labeled  with the absolute differences between the labels of their end-vertices, the n edges have the distinct labels 1,2,...,n. The gp/pari program below uses the Matrix-Tree Theorem and sums over {1,-1} vectors to isolate the multilinear term. It takes time 2^n * n^O(1) to compute graceful_tree_count(n). - Noam D. Elkies, Nov 18 2002 LINKS David Anick, Counting graceful labelings of trees: a theoretical and empirical study, Discrete Applied Mathematics 198 (2016), 65-81. FORMULA a(n) = 2 * A337274(n) for n >= 3. - Hugo Pfoertner, Oct 05 2020 EXAMPLE For n=3 we have 1-3-2 and 2-1-3, so a(3)=2. PROG (PARI) { treedet(v, n) = n=length(v); matdet(matrix(n, n, i, j, if(i-j, -v[abs(i-j)], sum(m=1, n+1, if(i-m, v[abs(i-m)], 0))))) } { graceful_tree_count(n, s, v, v1, v0)= if(n==1, 1, s=0; v1=vector(n-1, m, 1); v0=vector(n-1, m, if(m==1, 1, 0)); for(m=2^(n-2), 2^(n-1)-1, v= binary(m) - v0; s = s + (-1)^(v*v1~) * treedet(v1-2*v) ); s/2^(n-2) ) } \\ Noam D. Elkies, Nov 18 2002 for(n=1, 18, print1(graceful_tree_count(n), ", ")) \\ Example of function call CROSSREFS Cf. A006967, A337274. Sequence in context: A215070 A188479 A062962 * A134983 A330679 A218144 Adjacent sequences:  A033469 A033470 A033471 * A033473 A033474 A033475 KEYWORD nonn AUTHOR STATUS approved

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Last modified January 22 07:37 EST 2021. Contains 340360 sequences. (Running on oeis4.)