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A134484
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Triangle, read by rows, where T(n,k) = 2^[n*(n-1) - k*(k-1)] * C(n,k) for n>=k>=0.
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2
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1, 1, 1, 4, 8, 1, 64, 192, 48, 1, 4096, 16384, 6144, 256, 1, 1048576, 5242880, 2621440, 163840, 1280, 1, 1073741824, 6442450944, 4026531840, 335544320, 3932160, 6144, 1, 4398046511104, 30786325577728, 23089744183296, 2405181685760
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Has similar matrix power formulas as those for triangle A134049.
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FORMULA
| [T^(2^m)](n,k) = (2^m)^(n-k) * 2^[n*(n-1) - k*(k-1)] * C(n,k) for n>=k>=0 ; this is the formula for the matrix power T^(2^m) at row n and column k. Matrix log is given by: [log(T)](n+1,n) = (n+1)*4^n for n>=0 along a secondary diagonal with zeros elsewhere.
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EXAMPLE
| Matrix powers of triangle T also satisfy:
(1) [T^(2^m)](n,k) = T(n+m,k+m)/(2^m)^(n-k) for n>=k>=0;
(2) [T^( 1/2^(n-1) )](n,k) = (2^k)^(n-k) * C(n,k) for n>=k>=0;
compare to the formulas for matrix powers of triangle A134049.
Triangle T begins:
1;
1, 1;
4, 8, 1;
64, 192, 48, 1;
4096, 16384, 6144, 256, 1;
1048576, 5242880, 2621440, 163840, 1280, 1;
1073741824, 6442450944, 4026531840, 335544320, 3932160, 6144, 1; ...
Matrix log of triangle begins:
0;
1, 0;
0, 8, 0;
0, 0, 48, 0;
0, 0, 0, 256, 0;
0, 0, 0, 0, 1280, 0; ...
a single non-zero diagonal given by [log(T)](n+1,n) = (n+1)*4^n.
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PROG
| (PARI) {T(n, k)=2^(n*(n-1) - k*(k-1))*binomial(n, k)} (PARI) /* Matrix Power T^(2^m): */ {T(n, k, m)=2^(m*(n-k))*2^(n*(n-1) - k*(k-1))*binomial(n, k)}
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CROSSREFS
| Cf. A134049; A134485 (row sums).
Sequence in context: A021679 A197153 A196618 * A021958 A200412 A197483
Adjacent sequences: A134481 A134482 A134483 * A134485 A134486 A134487
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2007
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