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Search: id:A117250
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| A117250 |
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Triangle T, read by rows, where matrix power T^2 has powers of 2 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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+0
9
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| 1, 1, 1, -1, 2, 1, 4, -4, 4, 1, -40, 32, -16, 8, 1, 896, -640, 256, -64, 16, 1, -43008, 28672, -10240, 2048, -256, 32, 1, 4325376, -2752512, 917504, -163840, 16384, -1024, 64, 1, -899678208, 553648128, -176160768, 29360128, -2621440, 131072, -4096, 128, 1
(list; table; graph; listen)
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OFFSET
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0,5
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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) = A117251(n-k)*2^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-2*j)]/(n-k)!*2^(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;
-1,2,1;
4,-4,4,1;
-40,32,-16,8,1;
896,-640,256,-64,16,1;
-43008,28672,-10240,2048,-256,32,1;
4325376,-2752512,917504,-163840,16384,-1024,64,1;
-899678208,553648128,-176160768,29360128,-2621440,131072,-4096,128,1;
Matrix square T^2 has powers of 2 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. A117251 (column 0); variants: 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), A117265 (p=-2, q=2).
Adjacent sequences: A117247 A117248 A117249 this_sequence A117251 A117252 A117253
Sequence in context: A008741 A110316 A111975 this_sequence A136692 A101452 A019963
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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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