A338280 Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.

1, 3, 2, 16, 8, 3, 125, 50, 15, 4, 1296, 432, 108, 24, 5, 16807, 4802, 1029, 196, 35, 6, 262144, 65536, 12288, 2048, 320, 48, 7, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 2357947691, 428717762, 58461513, 7086244, 805255, 87846, 9317, 968, 99, 10

Offset

2,2

Comments

T(n, k) is the number of forests of n - k edges that connect every other labeled vertex to one of the k roots (see Section 3 in Wästlund).

References

Alfred Rényi, Some remarks on the theory of trees. MTA Mat. Kut. Inst. Kozl. (Publ. math. Inst. Hungar. Acad. Sci) 4 (1959), 73-85.

Links

Table of n, a(n) for n=2..56.

Arthur Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23: 376-378 (1889). Also in The collected mathematical papers of Arthur Cayley vol 13.

John Riordan, Forests of labeled trees, Journal of Combinatorial Theory 5 (1968), 93-103.

Lajos Takács, On Cayley’s Formula for Counting Forests, Journal of Combinatorial Theory Series A 53, 321-323 (1990). See Equation 1.

Johan Wästlund, Padlock Solitaire: A martingale trick for combinatorial enumeration, arXiv:2008.13017 [math.CO], 2020. See Section 3.

Mathematica

Table[k*n^(n-k-1), {n, 2, 11}, {k, 1, n-1}]//Flatten

Crossrefs

Cf. A000027 (diagonal), A000169, A000272 (1st column), A000312, A007334 (2nd column), A023811 (row sums), A034941, A072590, A075363, A210725.

Sequence in context: A292123 A084886 A275463 * A304989 A055864 A209600

Adjacent sequences: A338277 A338278 A338279 * A338281 A338282 A338283

Keyword

nonn,tabl

Author

Stefano Spezia, Oct 20 2020

Status

approved