|
%I #13 Oct 22 2020 01:45:30
%S 1,3,2,16,8,3,125,50,15,4,1296,432,108,24,5,16807,4802,1029,196,35,6,
%T 262144,65536,12288,2048,320,48,7,4782969,1062882,177147,26244,3645,
%U 486,63,8,100000000,20000000,3000000,400000,50000,6000,700,80,9,2357947691,428717762,58461513,7086244,805255,87846,9317,968,99,10
%N Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.
%C 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).
%D Alfred Rényi, Some remarks on the theory of trees. MTA Mat. Kut. Inst. Kozl. (Publ. math. Inst. Hungar. Acad. Sci) 4 (1959), 73-85.
%H Arthur Cayley, <a href="https://doi.org/10.1017/CBO9780511703799.010">A theorem on trees</a>, Quart. J. Pure Appl. Math. 23: 376-378 (1889). Also in The collected mathematical papers of Arthur Cayley vol 13.
%H John Riordan, <a href="https://doi.org/10.1016/S0021-9800(68)80033-X">Forests of labeled trees</a>, Journal of Combinatorial Theory 5 (1968), 93-103.
%H Lajos Takács, <a href="https://doi.org/10.1016/0097-3165(90)90064-4">On Cayley’s Formula for Counting Forests</a>, Journal of Combinatorial Theory Series A 53, 321-323 (1990). See Equation 1.
%H Johan Wästlund, <a href="https://arxiv.org/abs/2008.13017">Padlock Solitaire: A martingale trick for combinatorial enumeration</a>, arXiv:2008.13017 [math.CO], 2020. See Section 3.
%t Table[k*n^(n-k-1),{n,2,11},{k,1,n-1}]//Flatten
%Y Cf. A000027 (diagonal), A000169, A000272 (1st column), A000312, A007334 (2nd column), A023811 (row sums), A034941, A072590, A075363, A210725.
%K nonn,tabl
%O 2,2
%A _Stefano Spezia_, Oct 20 2020
|
