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 A241094 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. 4
 0, 1, 1, 4, 4, 4, 18, 24, 24, 18, 96, 144, 144, 96, 600, 960, 1080, 1080, 960, 600, 4320, 7200, 8460, 8460, 8460, 7200, 4320, 35280, 60840, 75600, 80640, 80640, 75600, 60480, 35280, 322560, 564480, 725760, 806400, 806400, 806400, 725760, 564480, 322560 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,4 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. LINKS Table of n, a(n) for n=2..45. C. Barrientos and S. M. Minion, Enumerating families of labeled graphs, J. Integer Seq., 18(2015), article 15.1.7. J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6. David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388. FORMULA 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). # alternative A241094 := proc(n,i) if n <2 or i<1 or i >= n then 0; elif i <= floor(n/2) then GAMMA(n-1)*(n-1-i)*i; else GAMMA(n-1)*(n-i)*(i-1) ; fi ; end proc: seq(seq(A241094(n,i),i=1..n-1),n=2..12); # R. J. Mathar, Jul 30 2024 EXAMPLE For n=7 and i=3, g(7,3) = 1080. For n=7 and i=5, g(7,5) = 960. Triangle begins: [n\i] [1] [2] [3] [4] [5] [6] [7] [8] [2] 0; [3] 1, 1; [4] 4, 4, 4; [5] 18, 24, 24, 18; [6] 96, 144, 144, 144, 96; [7] 600, 960, 1080, 1080, 960, 600; [8] 4320, 7200, 8640, 8640, 8640, 7200, 4320; [9] 35280, 60480, 75600, 80640, 80640, 75600, 60480, 35280; ... - Bruno Berselli, Apr 23 2014 MAPLE 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): MATHEMATICA n=10; (* This number must be replaced every time in order to produce the different entries of the sequence *) For[i = 1, i <= Floor[n/2], i++, g[n_, i_]:=(n-2)!*(n-1-i)*i; Print["g(", n, ", ", i, ")=", g[n, i]]] 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]]] PROG (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 CROSSREFS Cf. A001563, A003022, A004137, A005488, A006967, A033472, A081621, A103300, A117747, A212661. Sequence in context: A231700 A231746 A275858 * A319257 A131946 A034896 Adjacent sequences: A241091 A241092 A241093 * A241095 A241096 A241097 KEYWORD nonn,tabl,easy AUTHOR Christian Barrientos and Sarah Minion, Apr 15 2014 STATUS approved

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Last modified August 15 10:58 EDT 2024. Contains 375173 sequences. (Running on oeis4.)