A103242 - OEIS

%I #21 Jul 21 2024 00:28:05

%S 1,3,1,39,8,1,1206,176,15,1,69189,7784,495,24,1,6416568,585408,29430,

%T 1104,35,1,881032059,67481928,2791125,84600,2135,48,1,168514815360,

%U 11111547520,389244600,9841728,204470,3744,63,1,42934911510249

%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) = (1-(k+1)^2)^(n-k)/(n-k)! for n >= k >= 1.

%C 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).

%F For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(1-(m+1)^2)^(n-m)*T(m, k).

%F For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(1-(k+1)^2)^(j-k)*T(n, j).

%e Rows of unreduced fractions T(n,k)/(n-k)! begin:

%e   [1/0!],

%e   [3/1!, 1/0!],

%e   [39/2!, 8/1!, 1/0!],

%e   [1206/3!, 176/2!, 15/1!, 1/0!],

%e   [69189/4!, 7784/3!, 495/2!, 24/1!, 1/0!],

%e   [6416568/5!, 585408/4!, 29430/3!, 1104/2!, 35/1!, 1/0!], ...

%e forming the inverse of matrix P where P(n,k) = A103247(n,k)/(n-k)!:

%e   [1/0!],

%e   [ -3/1!, 1/0!],

%e   [9/2!, -8/1!, 1/0!],

%e   [ -27/3!, 64/2!, -15/1!, 1/0!],

%e   [81/4!, -512/3!, 225/2!, -24/1!, 1/0!],

%e   [ -243/5!, 4096/4!, -3375/3!, 576/2!, -35/1!, 1/0!], ...

%o (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)!))}

%Y Cf. A103247, A103236, A082171, A082163.

%K nonn,tabl,frac

%O 1,2

%A _Paul D. Hanna_, Feb 02 2005