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Search: id:A117262
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| A117262 |
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Triangle T, read by rows, where matrix inverse T^-1 has -3^n in the secondary diagonal: [T^-1](n+1,n) = -3^n, with all 1's in the main diagonal, and zeros elsewhere. |
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9
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| 1, 1, 1, 3, 3, 1, 27, 27, 9, 1, 729, 729, 243, 27, 1, 59049, 59049, 19683, 2187, 81, 1, 14348907, 14348907, 4782969, 531441, 19683, 243, 1, 10460353203, 10460353203, 3486784401, 387420489, 14348907, 177147, 729, 1
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=-1, q=3, r=1.
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FORMULA
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T(n,k) = 3^(n*(n-1)/2 - k*(k-1)/2).
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EXAMPLE
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Triangle T begins:
1;
1,1;
3,3,1;
27,27,9,1;
729,729,243,27,1;
59049,59049,19683,2187,81,1;
14348907,14348907,4782969,531441,19683,243,1;
10460353203,10460353203,3486784401,387420489,14348907,177147,729,1;
Matrix inverse T^-1 has -3^n in the 2nd diagonal:
1,
-1,1,
0,-3,1,
0,0,-9,1,
0,0,0,-27,1,
0,0,0,0,-81,1,
0,0,0,0,0,-243,1, ...
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PROGRAM
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(PARI) {T(n, k)=local(m=1, p=-1, q=3, r=1); prod(j=0, n-k-1, m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}
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CROSSREFS
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Cf. A047656 (column 0), A117263 (row sums); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117265 (p=-2, q=2).
Sequence in context: A121438 A108391 A111840 this_sequence A065431 A053375 A117252
Adjacent sequences: A117259 A117260 A117261 this_sequence A117263 A117264 A117265
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 14 2006
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