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.

1, 1, 1, 7, 4, 1, 142, 56, 9, 1, 5941, 1780, 207, 16, 1, 428856, 103392, 9342, 544, 25, 1, 47885899, 9649124, 709893, 32848, 1175, 36, 1, 7685040448, 1329514816, 82305144, 3142528, 91150, 2232, 49, 1, 1681740027657, 254821480596, 13598786979

Offset

1,4

Comments

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).

Links

Table of n, a(n) for n=1..39.

Formula

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

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

Example

Rows of unreduced fractions T(n,k)/(n-k)! begin:
  [1/0!],
  [1/1!, 1/0!],
  [7/2!, 4/1!, 1/0!],
  [142/3!, 56/2!, 9/1!, 1/0!],
  [5941/4!, 1780/3!, 207/2!, 16/1!, 1/0!],
  [428856/5!, 103392/4!, 9342/3!, 544/2!, 25/1!, 1/0!],
  [47885899/6!, 9649124/5!, 709893/4!, 32848/3!, 1175/2!, 36/1!, 1/0!], ...
forming the inverse of matrix P where P(n,k) = A103245(n,k)/(n-k)!:
  [1/0!],
  [-1/1!, 1/0!],
  [1/2!, -4/1!, 1/0!],
  [-1/3!, 16/2!, -9/1!, 1/0!],
  [1/4!, -64/3!, 81/2!, -16/1!, 1/0!], ...

Prog

(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<k||k<1, 0, (P^-1)[n, k]*(n-k)!))}

Crossrefs

Cf. A103245, A102086, A082169, A082157.

Sequence in context: A021907 A168422 A187056 * A155531 A216261 A188628

Adjacent sequences: A103237 A103238 A103239 * A103241 A103242 A103243

Keyword

nonn,tabl,frac

Author

Paul D. Hanna, Feb 02 2005

Status

approved