A071208 - OEIS

%I #39 Jun 29 2024 10:40:51

%S 1,2,2,6,15,6,24,104,104,24,120,770,1345,770,120,720,6264,16344,16344,

%T 6264,720,5040,56196,200452,300167,200452,56196,5040,40320,554112,

%U 2552192,5241984,5241984,2552192,554112,40320,362880,5973264,34138908,90857052,124756281,90857052,34138908,5973264,362880

%N Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the number k of decreasing edges.

%H Alois P. Heinz, <a href="/A071208/b071208.txt">Rows n = 1..141, flattened</a>

%H Cedric Chauve, S. Dulucq and O. Guibert, <a href="http://www.cecm.sfu.ca/~cchauve/Publications/SFCA00.ps">Enumeration of some labeled trees</a>, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.

%H Yen-Jen Cheng, Sen-Peng Eu, Tung-Shan Fu, and Jyun-Cheng Yao, <a href="https://arxiv.org/abs/2312.07351">On q-Counting of Noncrossing Chains and Parking Functions</a>, arXiv:2312.07351 [math.CO], 2023.

%H A. M. Khidr and B. S. El-Desouky, <a href="http://dx.doi.org/10.1016/S0195-6698(84)80018-9">A symmetric sum involving the Stirling numbers of the first kind</a>, European J. Combin., 5 (1984), 51-54. See Table 1.

%F T(n,k) = Sum_{m=k+1..n} (-1)^(k+1)*binomial(m,k+1)*Stirling1(n+1,n+1-m)*n^(n-m) with 0 <= k < n.

%e Triangle begins:

%e    1

%e    2    2

%e    6   15     6

%e   24  104   104    24

%e  120  770  1345   770   120

%e  720 6264 16344 16344  6264  720

%e  ...

%p T:= (n, k)-> add((-1)^(k+1)*binomial(m, k+1)*

%p              Stirling1(n+1, n+1-m)*n^(n-m), m=k+1..n):

%p seq(seq(T(n,k), k=0..n-1), n=1..9);

%t A071208[n_, k_] := Sum[(-1)^(k+1)*Binomial[m, k+1]*StirlingS1[n+1, n+1-m]*n^(n-m), {m, k+1, n}];

%t Table[A071208[n,k], {n, 10}, {k, 0, n-1}] (* _Paolo Xausa_, Jun 28 2024 *)

%Y Row sums give A000312.

%Y Cf. A000142.

%K easy,nonn,tabl

%O 1,2

%A Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

%E Offset 1 from _Alois P. Heinz_, Jun 26 2024