A103242 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.

1, 3, 1, 39, 8, 1, 1206, 176, 15, 1, 69189, 7784, 495, 24, 1, 6416568, 585408, 29430, 1104, 35, 1, 881032059, 67481928, 2791125, 84600, 2135, 48, 1, 168514815360, 11111547520, 389244600, 9841728, 204470, 3744, 63, 1, 42934911510249

Offset: 1

Author

Paul D. Hanna, Feb 02 2005

Keywords

• nonn
• tabl
• frac

Comments

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).

Examples

			Rows of unreduced fractions T(n,k)/(n-k)! begin:
  [1/0!],
  [3/1!, 1/0!],
  [39/2!, 8/1!, 1/0!],
  [1206/3!, 176/2!, 15/1!, 1/0!],
  [69189/4!, 7784/3!, 495/2!, 24/1!, 1/0!],
  [6416568/5!, 585408/4!, 29430/3!, 1104/2!, 35/1!, 1/0!], ...
forming the inverse of matrix P where P(n,k) = A103247(n,k)/(n-k)!:
  [1/0!],
  [ -3/1!, 1/0!],
  [9/2!, -8/1!, 1/0!],
  [ -27/3!, 64/2!, -15/1!, 1/0!],
  [81/4!, -512/3!, 225/2!, -24/1!, 1/0!],
  [ -243/5!, 4096/4!, -3375/3!, 576/2!, -35/1!, 1/0!], ...
		

Crossrefs

Cf. A103247, A103236, A082171, A082163.

Programs

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
				

Formula

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

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