0,4

Compare matrix power formulae to those of triangle A134484, where A134484(n,k) = 2^[n(n-1) - k(k-1)]*C(n,k).

The value of (2^m)-th matrix power of T at row n and column k is related to row n+m and column k+m of T by: [T^(2^m)](n,k) = T(n+m,k+m)/(2^m)^(n-k) for m>=0.

Below we illustrate this triangle and its 2 main properties:

(1) [T^(2^m)](n,k) = T(n+m,k+m)/(2^m)^(n-k) for m>=0;

(2) [T^( 1/2^(n-1) )](n,k) = (2^k)^(n-k) for n>=k>=0.

Triangle T begins:

1;

1, 1;

3, 4, 1;

23, 40, 16, 1;

512, 1072, 576, 64, 1;

34939, 84736, 56064, 8704, 256, 1;

7637688, 20930240, 16261120, 3190784, 135168, 1024, 1; ...

(1) Illustrate [T^(2^m)](n,k) = T(n+m,k+m)/(2^m)^(n-k) as follows.

Matrix square, T^2, begins:

1;

2, 1;

10, 8, 1;

134, 144, 32, 1;

5296, 7008, 2176, 128, 1; ...

where [T^(2^1)](n,k) = T(n+1,k+1)/2^(n-k).

Matrix 4-th power, T^4, begins:

1;

4, 1;

36, 16, 1;

876, 544, 64, 1;

63520, 49856, 8448, 256, 1; ...

where [T^(2^2)](n,k) = T(n+2,k+2)/4^(n-k).

Matrix 8-th power, T^8, begins:

1;

8, 1;

136, 32, 1;

6232, 2112, 128, 1;

854848, 374144, 33280, 512, 1; ...

where [T^(2^3)](n,k) = T(n+3,k+3)/8^(n-k).

(2) Illustrate [T^( 1/2^(n-1) )](n,k) = (2^k)^(n-k) as follows.

Matrix square-root, T^(1/2), begins:

1;

1/2, 1;

1, 2, 1; <== row 2: [T^(1/2^1)](2,k) = (2^k)^(2-k), k=0..2

9/2, 12, 8, 1; ...

Matrix 4-th root, T^(1/4), begins:

1;

1/4, 1;

3/8, 1, 1;

1, 4, 4, 1; <== row 3: [T^(1/2^2)](3,k) = (2^k)^(3-k), k=0..3

15/2, 36, 48, 16, 1; ...

Matrix 8-th root, T^(1/8), begins:

1;

1/8, 1;

5/32, 1/2, 1;

1/4, 3/2, 2, 1;

1, 8, 16, 8, 1; <== row 4: [T^(1/2^3)](4,k) = (2^k)^(4-k), k=0..4

107/8, 120, 288, 192, 32, 1; ...

(PARI) {T(n, k)=local(M=Mat(1), L, R); for(i=1, n, L=sum(j=1, #M, -(M^0-M)^j/j); M=sum(j=0, #L, (L/2^(#L-1))^j/j!); R=matrix(#M+1, #M+1, r, c, if(r>=c, if(r<=#M, M[r, c], 2^((c-1)*(#M+1-c))))); M=R^(2^(#M-1)) ); M[n+1, k+1]}

Cf. columns: A134050, A134051, A134052, A134053; A134054 (row sums).

Cf. A134484.

Sequence in context: A055133 A113084 A055325 this_sequence A157783 A123951 A123127

Adjacent sequences: A134046 A134047 A134048 this_sequence A134050 A134051 A134052

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 04 2007, Oct 28 2007