# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a103240 Showing 1-1 of 1 %I A103240 #21 Jul 21 2024 00:28:01 %S A103240 1,1,1,7,4,1,142,56,9,1,5941,1780,207,16,1,428856,103392,9342,544,25, %T A103240 1,47885899,9649124,709893,32848,1175,36,1,7685040448,1329514816, %U A103240 82305144,3142528,91150,2232,49,1,1681740027657,254821480596,13598786979 %N 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. %C A103240 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 A103240 For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2)^(n-m)*T(m, k). %F A103240 For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2)^(j-k)*T(n, j). %e A103240 Rows of unreduced fractions T(n,k)/(n-k)! begin: %e A103240 [1/0!], %e A103240 [1/1!, 1/0!], %e A103240 [7/2!, 4/1!, 1/0!], %e A103240 [142/3!, 56/2!, 9/1!, 1/0!], %e A103240 [5941/4!, 1780/3!, 207/2!, 16/1!, 1/0!], %e A103240 [428856/5!, 103392/4!, 9342/3!, 544/2!, 25/1!, 1/0!], %e A103240 [47885899/6!, 9649124/5!, 709893/4!, 32848/3!, 1175/2!, 36/1!, 1/0!], ... %e A103240 forming the inverse of matrix P where P(n,k) = A103245(n,k)/(n-k)!: %e A103240 [1/0!], %e A103240 [-1/1!, 1/0!], %e A103240 [1/2!, -4/1!, 1/0!], %e A103240 [-1/3!, 16/2!, -9/1!, 1/0!], %e A103240 [1/4!, -64/3!, 81/2!, -16/1!, 1/0!], ... %o A103240 (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