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A103243
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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) = (1-(k+1)^3)^(n-k)/(n-k)! for n>=k>=1.
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1
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1, 7, 1, 315, 26, 1, 45682, 2600, 63, 1, 15646589, 675194, 11655, 124, 1, 10567689552, 366349152, 4861458, 37944, 215, 1, 12503979423607, 361884843866, 3882676581, 23641468, 100835, 342, 1, 23841011541867520, 591934698991168, 5318920238688
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Define triangular matrix P by P(n,k) = (-k^3-3k^2-3k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103237 satisfies: M^3 + 3M^2 + 3M = 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 A082172 as a triangular matrix. The first column is A082160 (quasi-acyclic automata with 3 inputs).
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FORMULA
| For n>k>=1: 0 = Sum_{m=k..n} C(n-k, m-k)*(1-(m+1)^3)^(n-m)*T(m, k). For n>k>=1: 0 = Sum_{j=k..n} C(n-k, j-k)*(1-(k+1)^3)^(j-k)*T(n, j).
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EXAMPLE
| Rows of non-reduced fractions T(n,k)/(n-k)! begin:
[1/0! ],
[7/1!, 1/0! ],
[315/2!, 26/1!, 1/0! ],
[45682/3!, 2600/2!, 63/1!, 1/0! ],
[15646589/4!, 675194/3!, 11655/2!, 124/1!, 1/0! ],
[10567689552/5!,366349152/4!,4861458/3!,37944/2!,215/1!,1/0! ],...
forming the inverse of matrix P where P(n,k)=A103247(n,k)/(n-k)!:
[1/0! ],
[ -7/1!, 1/0! ],
[49/2!, -26/1!, 1/0! ],
[ -343/3!, 676/2!, -63/1!, 1/0! ],
[2401/4!, -17576/3!, 3969/2!, -124/1!, 1/0! ],
[ -16807/5!, 456976/4!, -250047/3!, 15376/2!, -215/1!, 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, (1-(c+1)^3)^(r-c)/(r-c)!))); return(if(n<k|k<1, 0, (P^-1)[n, k]*(n-k)!))}
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CROSSREFS
| Cf. A103248, A103237, A082172, A082160.
Sequence in context: A027538 A027478 A009792 * A027496 A056009 A159252
Adjacent sequences: A103240 A103241 A103242 * A103244 A103245 A103246
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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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