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Search: id:A117265
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| A117265 |
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Triangle T, read by rows, where matrix power T^-2 has -2^(n+1) in the secondary diagonal: [T^-2](n+1,n) = -2^(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, 2, 1, 20, 12, 4, 1, 280, 160, 48, 8, 1, 8064, 4480, 1280, 192, 16, 1, 473088, 258048, 71680, 10240, 768, 32, 1, 56229888, 30277632, 8257536, 1146880, 81920, 3072, 64, 1, 13495173120, 7197425664, 1937768448, 264241152, 18350080, 655360
(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=-2, q=2, r=1.
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FORMULA
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T(n,k) = A086229(n-k)*2^((n-k)*k). T(n,k) = 2^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(2*j+1) for n>k>=0, with T(n,n) = 1.
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EXAMPLE
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Triangle T begins:
1;
1,1;
3,2,1;
20,12,4,1;
280,160,48,8,1;
8064,4480,1280,192,16,1;
473088,258048,71680,10240,768,32,1;
56229888,30277632,8257536,1146880,81920,3072,64,1;
13495173120,7197425664,1937768448,264241152,18350080,655360,12288,128,1;
Matrix inverse square T^-2 has -2^(n+1) in the 2nd diagonal:
1;
-2,1;
0,-4,1;
0,0,-8,1;
0,0,0,-16,1;
0,0,0,0,-32,1;
0,0,0,0,0,-64,1; ...
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PROGRAM
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(PARI) {T(n, k)=local(m=1, p=-2, 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. A086229 (column 0), A117266 (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), A117262 (p=-1, q=3).
Sequence in context: A136733 A117269 A107862 this_sequence A107727 A087041 A144948
Adjacent sequences: A117262 A117263 A117264 this_sequence A117266 A117267 A117268
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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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