A134049 Triangle T, read by rows, where T(n,k) = [T^(2^k)](n-k,0) * (2^k)^(n-k) for n>=k>=0 such that row n of the 2^(n-1)-th root of T consists solely of integers given by: [T^( 1/2^(n-1) )](n,k) = (2^k)^(n-k) for n>=0.

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, 5539372954, 16855075840, 14918594560, 3501457408, 191561728, 2129920, 4096, 1, 13703105571256, 45696508860928, 45120522420224, 12230958252032, 813938245632, 11856248832, 33816576, 16384, 1, 118149647382446899, 427467706869837824, 463647865862488064, 141682892446105600, 11040640699727872, 197960679817216, 745898246144, 538968064, 65536, 1

Offset

0,4

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..495, of rows 0..30 of the flattened triangle.

Formula

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.

Example

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;
5539372954, 16855075840, 14918594560, 3501457408, 191561728, 2129920, 4096, 1;
13703105571256, 45696508860928, 45120522420224, 12230958252032, 813938245632, 11856248832, 33816576, 16384, 1;
118149647382446899, 427467706869837824, 463647865862488064, 141682892446105600, 11040640699727872, 197960679817216, 745898246144, 538968064, 65536, 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;
654070, 1016320, 398848, 33792, 512, 1; ...
where [T^(2^1)](n,k) = T(n+1,k+1)/2^(n-k).
Matrix 4th power, T^4, begins:
1;
4, 1;
36, 16, 1;
876, 544, 64, 1;
63520, 49856, 8448, 256, 1;
14568940, 13677568, 2993152, 133120, 1024, 1; ...
where [T^(2^2)](n,k) = T(n+2,k+2)/4^(n-k).
Matrix 8th power, T^8, begins:
1;
8, 1;
136, 32, 1;
6232, 2112, 128, 1;
854848, 374144, 33280, 512, 1;
373259224, 198715392, 23156736, 528384, 2048, 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;
58, 184, 160, 32, 1;
4475/2, 8192, 8576, 2304, 128, 1;
269828, 1118048, 1355776, 448512, 34816, 512, 1; ...
Matrix 4th 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;
667/4, 928, 1472, 640, 64, 1;
11180, 71600, 131072, 68608, 9216, 256, 1; ...
Matrix 8th 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;
977/2, 5336, 14848, 11776, 2560, 128, 1; ...
Matrix 16th root, T^(1/8), begins:
1;
1/16, 1;
9/128, 1/4, 1;
9/128, 5/8, 1, 1;
11/128, 2, 6, 4, 1;
1, 16, 64, 64, 16, 1; <== row 5: [T^(1/2^4)](5,k) = (2^k)^(5-k), k=0..5
139/8, 428, 1920, 2304, 768, 64, 1; ...

Prog

(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]}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

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

Cf. A134484.

Cf. A274477 (matrix log).

Sequence in context: A255905 A055325 A162498 * A224069 A157783 A123951

Adjacent sequences: A134046 A134047 A134048 * A134050 A134051 A134052

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 04 2007, Oct 28 2007

Status

approved