A393321 Triangle read by rows: T(n,k) is the number of graceful Prüfer codes on n vertices whose first element equals k (0 <= k <= n-1).

1, 0, 1, 1, 0, 1, 2, 2, 1, 2, 1, 6, 5, 4, 4, 4, 5, 18, 18, 15, 13, 12, 14, 25, 67, 63, 62, 47, 47, 55, 76, 101, 301

Offset

3,7

Comments

Any labeled tree on n vertices admits a unique Prüfer code. A graceful labeling of a tree on n vertices assigns the labels {0, 1, ..., n-1} to the vertices such that the induced edge labels (absolute differences of endpoint labels) are exactly {1, 2, ..., n-1}. This triangle, read by rows, gives T(n,k), the number of Prüfer codes of gracefully labeled trees on n vertices whose first element equals k, where 0 <= k <= n-1.

References

Douglas B. West, Introduction to Graph Theory, Pearson Education Pte. Ltd, 2002, page 81.

Links

Igor Blokhin, Table of n, a(n) for n = 3..104

Igor Blokhin, Graceful Prüfer codes (Python repository).

Example

For n=4 the row is [1,0,1,2], meaning that among all graceful Prüfer codes (of length 2) on vertices {0,1,2,3}, there are 1 code starting in 0, no codes starting in 1, 1 code starting in 2, and 2 code starting in 3.
Triangle T(n,k), 0 <= k <= n-1 (each row sums to A033472(n)):
  n=3:    1    0    1
  n=4:    1    0    1    2
  n=5:    2    1    2    1    6
  n=6:    5    4    4    4    5   18
  n=7:   18   15   13   12   14   25    67
  n=8:   63   62   47   47   55   76   101   301

Crossrefs

Cf. A033472, A392082, A392515, A392516.

Sequence in context: A030768 A051480 A071572 * A172176 A285930 A300480

Adjacent sequences: A393318 A393319 A393320 * A393322 A393323 A393324

Keyword

nonn,hard,more,tabf

Author

Igor Blokhin, Feb 10 2026

Status

approved