OFFSET
1,2
COMMENTS
Define a triangular matrix P where P(n,k) = (-k^2-2*k)^(n-k)/(n-k)!; then M = P*D*P^-1 = A103236 satisfies M^2 + 2*M = 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 A082171 as a triangular matrix. The first column is A082163 (enumerates acyclic automata with 2 inputs).
FORMULA
For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(1-(m+1)^2)^(n-m)*T(m, k).
For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(1-(k+1)^2)^(j-k)*T(n, j).
EXAMPLE
Rows of unreduced fractions T(n,k)/(n-k)! begin:
[1/0!],
[3/1!, 1/0!],
[39/2!, 8/1!, 1/0!],
[1206/3!, 176/2!, 15/1!, 1/0!],
[69189/4!, 7784/3!, 495/2!, 24/1!, 1/0!],
[6416568/5!, 585408/4!, 29430/3!, 1104/2!, 35/1!, 1/0!], ...
forming the inverse of matrix P where P(n,k) = A103247(n,k)/(n-k)!:
[1/0!],
[ -3/1!, 1/0!],
[9/2!, -8/1!, 1/0!],
[ -27/3!, 64/2!, -15/1!, 1/0!],
[81/4!, -512/3!, 225/2!, -24/1!, 1/0!],
[ -243/5!, 4096/4!, -3375/3!, 576/2!, -35/1!, 1/0!], ...
PROG
(PARI) {T(n, k)=my(P); if(n>=k&k>=1, P=matrix(n, n, r, c, if(r>=c, (1-(c+1)^2)^(r-c)/(r-c)!))); return(if(n<k||k<1, 0, (P^-1)[n, k]*(n-k)!))}
CROSSREFS
KEYWORD
AUTHOR
Paul D. Hanna, Feb 02 2005
STATUS
approved