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.

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

Offset: 1

Author

Paul D. Hanna, Feb 02 2005

Keywords

• nonn
• tabl
• frac

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).

Examples

			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!], ...
		

Crossrefs

Cf. A103246, A102098, A082170, A082162.

Programs

PARI

{T(n,k)=my(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(-c^3)^(r-c)/(r-c)!))); return(if(n
				

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).