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 A103241 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^3)^(n-k)/(n-k)! for n >= k >= 1. 0
 1, 1, 1, 15, 8, 1, 1024, 368, 27, 1, 198581, 53672, 2727, 64, 1, 85102056, 18417792, 710532, 11904, 125, 1, 68999174203, 12448430408, 386023509, 4975936, 38375, 216, 1, 95264160938080, 14734002979456, 381535651512, 3977848832, 23945000 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Define a triangular matrix P where P(n,k) = (-k^3)^(n-k)/(n-k)!, then M = P*D*P^-1 = A102098 satisfies M^3 = 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 A082170 as a triangular matrix. The first column is A082162 (enumerates acyclic automata with 3 inputs). LINKS FORMULA For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^3)^(n-m)*T(m, k). For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^3)^(j-k)*T(n, j). EXAMPLE Rows of unreduced fractions T(n,k)/(n-k)! begin:   [1/0!],   [1/1!, 1/0!],   [15/2!, 8/1!, 1/0!],   [1024/3!, 368/2!, 27/1!, 1/0!],   [198581/4!, 53672/3!, 2727/2!, 64/1!, 1/0!],   [85102056/5!, 18417792/4!, 710532/3!, 11904/2!, 125/1!, 1/0!], ... forming the inverse of matrix P where P(n,k) = A103246(n,k)/(n-k)!:   [1/0!],   [-1/1!, 1/0!],   [1/2!, -8/1!, 1/0!],   [-1/3!, 64/2!, -27/1!, 1/0!],   [1/4!, -512/3!, 729/2!, -64/1!, 1/0!], ... PROG (PARI) {T(n, k)=local(P); if(n>=k&k>=1, P=matrix(n, n, r, c, if(r>=c, (-c^3)^(r-c)/(r-c)!))); return(if(n

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Last modified August 4 18:15 EDT 2021. Contains 346455 sequences. (Running on oeis4.)