%I #21 Jul 21 2024 00:28:01
%S 1,1,1,7,4,1,142,56,9,1,5941,1780,207,16,1,428856,103392,9342,544,25,
%T 1,47885899,9649124,709893,32848,1175,36,1,7685040448,1329514816,
%U 82305144,3142528,91150,2232,49,1,1681740027657,254821480596,13598786979
%N 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.
%C 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).
%F For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2)^(n-m)*T(m, k).
%F For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2)^(j-k)*T(n, j).
%e Rows of unreduced fractions T(n,k)/(n-k)! begin:
%e [1/0!],
%e [1/1!, 1/0!],
%e [7/2!, 4/1!, 1/0!],
%e [142/3!, 56/2!, 9/1!, 1/0!],
%e [5941/4!, 1780/3!, 207/2!, 16/1!, 1/0!],
%e [428856/5!, 103392/4!, 9342/3!, 544/2!, 25/1!, 1/0!],
%e [47885899/6!, 9649124/5!, 709893/4!, 32848/3!, 1175/2!, 36/1!, 1/0!], ...
%e forming the inverse of matrix P where P(n,k) = A103245(n,k)/(n-k)!:
%e [1/0!],
%e [-1/1!, 1/0!],
%e [1/2!, -4/1!, 1/0!],
%e [-1/3!, 16/2!, -9/1!, 1/0!],
%e [1/4!, -64/3!, 81/2!, -16/1!, 1/0!], ...
%o (PARI) {T(n,k)=my(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)!))}
%Y Cf. A103245, A102086, A082169, A082157.
%K nonn,tabl,frac
%O 1,4
%A _Paul D. Hanna_, Feb 02 2005