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Search: id:A117258
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| A117258 |
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Triangle T, read by rows, where matrix power T^2 has 2*4^n in the secondary diagonal: [T^2](n+1,n) = 2*4^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, 4, 1, 32, -32, 16, 1, -2560, 2048, -512, 64, 1, 917504, -655360, 131072, -8192, 256, 1, -1409286144, 939524096, -167772160, 8388608, -131072, 1024, 1, 9070970929152, -5772436045824, 962072674304, -42949672960, 536870912, -2097152, 4096, 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=2, q=4, r=1.
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FORMULA
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T(n,k) = A117259(n-k)*4^((n-k)*k). T(n,k) = (-1)^(n-k)*4^(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;
-2,4,1;
32,-32,16,1;
-2560,2048,-512,64,1;
917504,-655360,131072,-8192,256,1;
-1409286144,939524096,-167772160,8388608,-131072,1024,1;
Matrix square T^2 has 2*4^n in the 2nd diagonal:
1,
2,1,
0,8,1,
0,0,32,1,
0,0,0,128,1,
0,0,0,0,512,1, ...
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PROGRAM
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(PARI) {T(n, k)=local(m=1, p=2, q=4, 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. A117259 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (q=5), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).
Sequence in context: A117131 A030043 A045497 this_sequence A009417 A009451 A077748
Adjacent sequences: A117255 A117256 A117257 this_sequence A117259 A117260 A117261
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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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