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Search: id:A117252
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| A117252 |
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Triangle T, read by rows, where matrix power T^3 has powers of 3 in the secondary diagonal: [T^3](n+1,n) = 3^(n+1), 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, 45, -27, 9, 1, -2430, 1215, -243, 27, 1, 433026, -196830, 32805, -2187, 81, 1, -245525742, 105225318, -15943230, 885735, -19683, 243, 1, 434685788658, -178988265918, 25569752274, -1291401630, 23914845, -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=3, q=3, r=1.
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FORMULA
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T(n,k) = A117253(n-k)*3^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-3*j)]/(n-k)!*3^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1.
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EXAMPLE
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Triangle T begins:
1;
1,1;
-3,3,1;
45,-27,9,1;
-2430,1215,-243,27,1;
433026,-196830,32805,-2187,81,1;
-245525742,105225318,-15943230,885735,-19683,243,1;
434685788658,-178988265918,25569752274,-1291401630,23914845,-177147,729,1;
Matrix cube T^3 has powers of 3 in the 2nd diagonal:
1;
3,1;
0,9,1;
0,0,27,1;
0,0,0,81,1;
0,0,0,0,243,1;
0,0,0,0,0,729,1; ...
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PROGRAM
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(PARI) {T(n, k)=local(m=1, p=3, 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. A117253 (column 0); variants: A117250 (p=q=2), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).
Sequence in context: A117262 A065431 A053375 this_sequence A016037 A106449 A097278
Adjacent sequences: A117249 A117250 A117251 this_sequence A117253 A117254 A117255
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KEYWORD
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sign,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 14 2006
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