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%I #16 Jan 12 2024 11:56:39
%S 1,2,2,9,6,12,64,36,48,108,625,320,360,540,1280,7776,3750,3840,4860,
%T 7680,18750,117649,54432,52500,60480,80640,131250,326592,2097152,
%U 941192,870912,945000,1146880,1575000,2612736,6588344,43046721
%N Triangle read by rows: T(n, k) = binomial(n, k)*k^k*(n-k)^(n-k-1) k=0..n-1.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 68 (2.1.43).
%F E.g.f.: -LambertW(-y)/(1+LambertW(-x*y)). - _Vladeta Jovovic_, Jan 26 2006
%F T(n, k) = n*binomial(n-1, k-1)*(k-1)^(k-1)*(n-k+1)^(n-k-1) assuming offset (1, 1). - _Peter Luschny_, Jan 12 2024
%e Triangle starts:
%e [1][ 1]
%e [2][ 2, 2]
%e [3][ 9, 6, 12]
%e [4][ 64, 36, 48, 108]
%e [5][ 625, 320, 360, 540, 1280]
%e [6][ 7776, 3750, 3840, 4860, 7680, 18750]
%e [7][ 117649, 54432, 52500, 60480, 80640, 131250, 326592]
%e [8][2097152, 941192, 870912, 945000, 1146880, 1575000, 2612736, 6588344]
%o (Julia) # Assuming offset (n=1, k=1).
%o T(n, k) = binomial(n-1, k-1)*(k-1)^(k-1)*n*(n-k+1)^(n-k-1)
%o for n in 1:9 (println([n], [T(n, k) for k in 1:n])) end
%o # _Peter Luschny_, Jan 12 2024
%Y T = n * A185390 after proper alignment of offsets.
%Y Columns 1, 2: A000169, A055541.
%Y Main diagonal: A055897.
%Y Row sums give A000312.
%K nonn,tabl
%O 1,2
%A _Christian G. Bower_, Dec 13 2001