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Search: id:A117260
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| A117260 |
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Triangle T, read by rows, where matrix inverse T^-1 has -2^n in the secondary diagonal: [T^-1](n+1,n) = -2^n, with all 1's in the main diagonal, and zeros elsewhere. |
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9
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| 1, 1, 1, 2, 2, 1, 8, 8, 4, 1, 64, 64, 32, 8, 1, 1024, 1024, 512, 128, 16, 1, 32768, 32768, 16384, 4096, 512, 32, 1, 2097152, 2097152, 1048576, 262144, 32768, 2048, 64, 1, 268435456, 268435456, 134217728, 33554432, 4194304, 262144, 8192, 128, 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=2, r=1.
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FORMULA
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T(n,k) = 2^(n*(n-1)/2 - k*(k-1)/2).
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EXAMPLE
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Triangle T begins:
1;
1,1;
2,2,1;
8,8,4,1;
64,64,32,8,1;
1024,1024,512,128,16,1;
32768,32768,16384,4096,512,32,1;
2097152,2097152,1048576,262144,32768,2048,64,1;
268435456,268435456,134217728,33554432,4194304,262144,8192,128,1;
Matrix inverse T^-1 has -2^n in the 2nd diagonal:
1,
-1,1,
0,-2,1,
0,0,-4,1,
0,0,0,-8,1,
0,0,0,0,-16,1,
0,0,0,0,0,-32,1, ...
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PROGRAM
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(PARI) {T(n, k)=local(m=1, p=-1, q=2, 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. A006125 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117262 (p=-1, q=3), A117265 (p=-2, q=2).
Sequence in context: A096440 A121350 A135080 this_sequence A077944 A077992 A126586
Adjacent sequences: A117257 A117258 A117259 this_sequence A117261 A117262 A117263
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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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