0,4

Has similar matrix power formulas as those for triangle A134049.

[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.

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.

(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)}

Cf. A134049; A134485 (row sums).

Sequence in context: A002390 A122149 A021679 this_sequence A021958 A019954 A072616

Adjacent sequences: A134481 A134482 A134483 this_sequence A134485 A134486 A134487

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2007