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A103240
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Non-reduced 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.
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0
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Define triangular matrix P by 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).
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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).
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EXAMPLE
| Rows of non-reduced 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! ],...
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PROG
| (PARI) {T(n, k)=local(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)!))}
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CROSSREFS
| Cf. A103245, A102086, A082169, A082157.
Sequence in context: A021907 A168422 A187056 * A155531 A188628 A021578
Adjacent sequences: A103237 A103238 A103239 * A103241 A103242 A103243
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KEYWORD
| nonn,tabl,frac
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Feb 02 2005
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