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A103244
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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-k)^(n-k)/(n-k)! for n>=k>=1.
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2
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1, 2, 1, 20, 6, 1, 512, 108, 12, 1, 25392, 4104, 336, 20, 1, 2093472, 273456, 17568, 800, 30, 1, 260555392, 28515456, 1500288, 54800, 1620, 42, 1, 45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1, 10849051434240, 894918533760
(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^2-k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103238 satisfies: M^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. First column is A103353.
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FORMULA
| For n>k>=1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2-m)^(n-m)*T(m, k). For n>k>=1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2-k)^(j-k)*T(n, j).
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EXAMPLE
| Rows of non-reduced fractions T(n,k)/(n-k)! begin:
[1/0! ],
[2/1!, 1/0! ],
[20/2!, 6/1!, 1/0! ],
[512/3!, 108/2!, 12/1!, 1/0! ],
[25392/4!, 4104/3!, 336/2!, 20/1!, 1/0! ],
[2093472/5!, 273456/4!, 17568/3!, 800/2!, 30/1!, 1/0! ],...
forming the inverse of matrix P where P(n,k)=A103249(n,k)/(n-k)!:
[1/0! ],
[ -2/1!, 1/0! ],
[4/2!, -6/1!, 1/0! ],
[ -8/3!, 36/2!, -12/1!, 1/0! ],
[16/4!, -216/3!, 144/2!, -20/1!, 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-c)^(r-c)/(r-c)!))); return(if(n<k|k<1, 0, (P^-1)[n, k]*(n-k)!))}
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CROSSREFS
| Cf. A103249, A103238, A103353.
Sequence in context: A013021 A012907 A066753 * A185169 A012927 A013158
Adjacent sequences: A103241 A103242 A103243 * A103245 A103246 A103247
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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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