A066320 Triangle read by rows: T(n, k) = binomial(n, k)*k^k*(n-k)^(n-k-1) k=0..n-1.

1, 2, 2, 9, 6, 12, 64, 36, 48, 108, 625, 320, 360, 540, 1280, 7776, 3750, 3840, 4860, 7680, 18750, 117649, 54432, 52500, 60480, 80640, 131250, 326592, 2097152, 941192, 870912, 945000, 1146880, 1575000, 2612736, 6588344, 43046721

Offset: 1

Author

Christian G. Bower, Dec 13 2001

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  [1][      1]
  [2][      2,      2]
  [3][      9,      6,     12]
  [4][     64,     36,     48,    108]
  [5][    625,    320,    360,    540,    1280]
  [6][   7776,   3750,   3840,   4860,    7680,   18750]
  [7][ 117649,  54432,  52500,  60480,   80640,  131250,  326592]
  [8][2097152, 941192, 870912, 945000, 1146880, 1575000, 2612736, 6588344]
		

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 68 (2.1.43).

Crossrefs

T = n * A185390 after proper alignment of offsets.

Columns 1, 2: A000169, A055541.

Main diagonal: A055897.

Row sums give A000312.

Programs

Julia

# Assuming offset (n=1, k=1).
T(n, k) = binomial(n-1, k-1)*(k-1)^(k-1)*n*(n-k+1)^(n-k-1)
for n in 1:9 (println([n], [T(n, k) for k in 1:n])) end
# Peter Luschny, Jan 12 2024

Formula

E.g.f.: -LambertW(-y)/(1+LambertW(-x*y)). - Vladeta Jovovic, Jan 26 2006

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