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Irregular triangle read by rows: T(n,k) = total number of graceful labelings of connected graphs with n edges (n>=1) and k vertices (k>=1).
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%I #15 Dec 02 2019 06:15:09

%S 0,1,0,0,2,0,0,2,4,0,0,0,12,12,0,0,0,8,68,40,0,0,0,2,106,400,164,0,0,

%T 0,0,88,1184,2496,752,0,0,0,0,32,1728,11932,17338,4020,0,0,0,0,8,1696,

%U 29444,121084,127660,23576,0,0,0,0,0,964,45472,442222,1259838,1030524,155632,0,0,0,0,0,432,50292,1051048,6488844,13590300,8852398,1112032

%N Irregular triangle read by rows: T(n,k) = total number of graceful labelings of connected graphs with n edges (n>=1) and k vertices (k>=1).

%C Consider a connected graph with E edges and V vertices. The vertices are given labels in the range 0 to E so that the differences between edges' endpoints are {1,...,E}. None of the vertices are isolated; hence each vertex label participates in at least one edge.

%H D. E. Knuth, <a href="https://cs.stanford.edu/~knuth/programs/graceful-count.w">Program for counting graceful graph labelings.</a>

%H D. E. Knuth, <a href="/A329790/a329790.txt">Output from program</a>

%e Rows 1 through 16 are:

%e 0,1,

%e 0,0,2,

%e 0,0,2,4,

%e 0,0,0,12,12,

%e 0,0,0,8,68,40,

%e 0,0,0,2,106,400,164,

%e 0,0,0,0,88,1184,2496,752,

%e 0,0,0,0,32,1728,11932,17338,4020,

%e 0,0,0,0,8,1696,29444,121084,127660,23576,

%e 0,0,0,0,0,964,45472,442222,1259838,1030524,155632,

%e 0,0,0,0,0,432,50292,1051048,6488844,13590300,8852398,1112032,

%e 0,0,0,0,0,104,40176,1744982,21410862,93887854,153385084,82121018,8733628,

%e 0,0,0,0,0,24,24688,2250864,51470648,420080516,1378054500,1803091192,806839236,73547332,

%e 0,0,0,0,0,0,11208,2184056,92513144,1340857860,7975757616,20604457050,22263918324,8481362264,670789524,

%e 0,0,0,0,0,0,3780,1818244,136668880,3335302886,33380409234,151167588580,315541333316,286284099998,93933923996,6502948232,

%e 0,0,0,0,0,0,864,1141312,159551124,6537511962,106698003000,795992914532,2869123162654,4974721374674,3859250594040,1104325114202,67540932632,

%Y A033472, A329788, A329789 are diagonals.

%K nonn,tabf

%O 1,5

%A _N. J. A. Sloane_, Dec 02 2019, based on email from _Don Knuth_, Dec 01 2019