|
| |
|
|
A103240
|
|
Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^2)^(n-k)/(n-k)! for n >= k >= 1.
|
|
0
|
|
|
|
1, 1, 1, 7, 4, 1, 142, 56, 9, 1, 5941, 1780, 207, 16, 1, 428856, 103392, 9342, 544, 25, 1, 47885899, 9649124, 709893, 32848, 1175, 36, 1, 7685040448, 1329514816, 82305144, 3142528, 91150, 2232, 49, 1, 1681740027657, 254821480596, 13598786979
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,4
|
|
|
COMMENTS
|
Define a triangular matrix P where P(n,k) = (-k^2)^(n-k)/(n-k)!; then M = P*D*P^-1 = A102086 satisfies M^2 = SHIFTUP(M) where D is the diagonal matrix consisting of {1,2,3,...}. The operation SHIFTUP(M) shifts each column of M up 1 row. Essentially equal to square array A082169 as a triangular matrix. The first column is A082157 (enumerates acyclic automata with 2 inputs).
|
|
|
LINKS
|
|
|
|
FORMULA
|
For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2)^(n-m)*T(m, k).
For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2)^(j-k)*T(n, j).
|
|
|
EXAMPLE
|
Rows of unreduced fractions T(n,k)/(n-k)! begin:
[1/0!],
[1/1!, 1/0!],
[7/2!, 4/1!, 1/0!],
[142/3!, 56/2!, 9/1!, 1/0!],
[5941/4!, 1780/3!, 207/2!, 16/1!, 1/0!],
[428856/5!, 103392/4!, 9342/3!, 544/2!, 25/1!, 1/0!],
[47885899/6!, 9649124/5!, 709893/4!, 32848/3!, 1175/2!, 36/1!, 1/0!], ...
forming the inverse of matrix P where P(n,k) = A103245(n,k)/(n-k)!:
[1/0!],
[-1/1!, 1/0!],
[1/2!, -4/1!, 1/0!],
[-1/3!, 16/2!, -9/1!, 1/0!],
[1/4!, -64/3!, 81/2!, -16/1!, 1/0!], ...
|
|
|
PROG
|
(PARI) {T(n, k)=local(P); if(n>=k&k>=1, P=matrix(n, n, r, c, if(r>=c, (-c^2)^(r-c)/(r-c)!))); return(if(n<k|k<1, 0, (P^-1)[n, k]*(n-k)!))}
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
