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%I #48 Sep 08 2022 08:46:07
%S 0,1,1,4,4,4,18,24,24,18,96,144,144,96,600,960,1080,1080,960,600,4320,
%T 7200,8460,8460,8460,7200,4320,35280,60840,75600,80640,80640,75600,
%U 60480,35280,322560,564480,725760,806400,806400,806400,725760,564480,322560
%N Triangle read by rows: T(n,i) = number of gracefully labeled graphs with n edges that do not use the label i, 1 <= i <= n-1, n > 1.
%C 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.
%H C. Barrientos and S. M. Minion, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Minion/minion3.pdf">Enumerating families of labeled graphs</a>, J. Integer Seq., 18(2015), article 15.1.7.
%H J. A. Gallian, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6">A dynamic survey of graph labeling</a>, Elec. J. Combin., (2013), #DS6.
%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., 15(1976), no. 4, 379-388.
%F For n >=2, if 1 <= i <= floor(n/2), g(n,i) = (n-2)!*(n-1-i)*i; if ceiling((n+1)/2) <= i <= n-1, g(n,i) = (n-2)!*(n-i)*(i-1).
%e For n=7 and i=3, g(7,3) = 1080.
%e For n=7 and i=5, g(7,5) = 960.
%e Triangle begins:
%e [n\i] [1] [2] [3] [4] [5] [6] [7] [8]
%e [2] 0;
%e [3] 1, 1;
%e [4] 4, 4, 4;
%e [5] 18, 24, 24, 18;
%e [6] 96, 144, 144, 144, 96;
%e [7] 600, 960, 1080, 1080, 960, 600;
%e [8] 4320, 7200, 8640, 8640, 8640, 7200, 4320;
%e [9] 35280, 60480, 75600, 80640, 80640, 75600, 60480, 35280;
%e ...
%e - _Bruno Berselli_, Apr 23 2014
%p Labeled:=(i,n) piecewise(n<2 or i<1, -infinity, 1 <= i <= floor(n/2), GAMMA(n-1)*(n-1-i)*i, ceil((n+1)/2) <= i <= n-1, GAMMA(n-1)*(n-i)*(i-1), infinity):
%t n=10; (* This number must be replaced every time in order to produce the different entries of the sequence *)
%t For[i = 1, i <= Floor[n/2], i++, g[n_,i_]:=(n-2)!*(n-1-i)*i; Print["g(",n,",",i,")=", g[n,i]]]
%t For[i = Ceiling[(n+1)/2], i <= (n-1), i++, g[n_,i_]:=(n-2)!*(n-i)*(i-1); Print["g(",n,",",i,")=",g[n,i]]]
%o (Magma) /* As triangle: */ [[i le Floor(n/2) select Factorial(n-2)*(n-1-i)*i else Factorial(n-2)*(n-i)*(i-1): i in [1..n-1]]: n in [2..10]]; // _Bruno Berselli_, Apr 23 2014
%Y Cf. A001563, A003022, A004137, A005488, A006967, A033472, A081621, A103300, A117747, A212661.
%K nonn,tabl,easy
%O 2,4
%A _Christian Barrientos_ and _Sarah Minion_, Apr 15 2014
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