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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. 3
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).

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 December 3 13:43 EST 2021. Contains 349463 sequences. (Running on oeis4.)