A223257 Triangle read by rows: T(0,0)=1; for n>=1 T(n,k) is the denominator of the coefficient of x^k in the characteristic polynomial of the matrix realizing the transformation to Jacobi coordinates for a system of n particles on a line.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 24, 12, 1, 1, 60, 120, 120, 60, 1, 1, 20, 180, 720, 180, 20, 1, 1, 140, 126, 1680, 1680, 126, 140, 1, 1, 280, 10080, 10080, 40320, 10080, 10080, 280, 1, 1, 2520, 10080, 1296, 3456, 3456, 1296, 10080, 2520, 1

Offset: 0

Author

Alberto Tacchella, Mar 18 2013

Keywords

• easy
• frac
• nonn
• tabl

Comments

The matrix J(n) realizing the change of coordinates for n particles is

[1, -1, 0, 0, 0, ... 0],

[1/2, 1/2, -1, 0, ... 0],

[1/3, 1/3, 1/3, -1, 0 ... 0],

...

[1/n, 1/n, 1/n, 1/n, ... 1/n]

Diagonals T(n,1)=T(n,n-1) are A002805, corresponding to the fact that the matrix J(n) above has trace equal to the n-th harmonic number.

See A223256 for numerators.

Examples

			Triangle begins:
1,
1, 1,
1, 2, 1,
1, 6, 6, 1,
1, 12, 24, 12, 1,
1, 60, 120, 120, 60, 1,
1, 20, 180, 720, 180, 20, 1,
1, 140, 126, 1680, 1680, 126, 140, 1,
...
		

Links

• Wikipedia, Jacobi coordinates