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A103241
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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^3)^(n-k)/(n-k)! for n>=k>=1.
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0
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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; internal format)
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OFFSET
| 1,4
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COMMENTS
| Define triangular matrix P by 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).
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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).
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EXAMPLE
| Rows of non-reduced 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! ],...
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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^3)^(r-c)/(r-c)!))); return(if(n<k|k<1, 0, (P^-1)[n, k]*(n-k)!))}
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CROSSREFS
| Cf. A103246, A102098, A082170, A082162.
Sequence in context: A168211 A131876 A126070 * A194707 A168548 A094501
Adjacent sequences: A103238 A103239 A103240 * A103242 A103243 A103244
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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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