A329790 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).

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

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..77.

D. E. Knuth, Program for counting graceful graph labelings.

D. E. Knuth, Output from program

Example

Rows 1 through 16 are:
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,
0,0,0,0,88,1184,2496,752,
0,0,0,0,32,1728,11932,17338,4020,
0,0,0,0,8,1696,29444,121084,127660,23576,
0,0,0,0,0,964,45472,442222,1259838,1030524,155632,
0,0,0,0,0,432,50292,1051048,6488844,13590300,8852398,1112032,
0,0,0,0,0,104,40176,1744982,21410862,93887854,153385084,82121018,8733628,
0,0,0,0,0,24,24688,2250864,51470648,420080516,1378054500,1803091192,806839236,73547332,
0,0,0,0,0,0,11208,2184056,92513144,1340857860,7975757616,20604457050,22263918324,8481362264,670789524,
0,0,0,0,0,0,3780,1818244,136668880,3335302886,33380409234,151167588580,315541333316,286284099998,93933923996,6502948232,
0,0,0,0,0,0,864,1141312,159551124,6537511962,106698003000,795992914532,2869123162654,4974721374674,3859250594040,1104325114202,67540932632,

Crossrefs

A033472, A329788, A329789 are diagonals.

Sequence in context: A115509 A279360 A134312 * A343649 A195581 A020474

Adjacent sequences: A329787 A329788 A329789 * A329791 A329792 A329793

Keyword

nonn,tabf

Author

N. J. A. Sloane, Dec 02 2019, based on email from Don Knuth, Dec 01 2019

Status

approved