A134049 - OEIS

%I #23 Nov 13 2016 15:46:06

%S 1,1,1,3,4,1,23,40,16,1,512,1072,576,64,1,34939,84736,56064,8704,256,

%T 1,7637688,20930240,16261120,3190784,135168,1024,1,5539372954,

%U 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

%N 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.

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

%H Paul D. Hanna, <a href="/A134049/b134049.txt">Table of n, a(n) for n = 0..495, of rows 0..30 of the flattened triangle.</a>

%F 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.

%e Below we illustrate this triangle and its 2 main properties:

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

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

%e Triangle T begins:

%e 1;

%e 1, 1;

%e 3, 4, 1;

%e 23, 40, 16, 1;

%e 512, 1072, 576, 64, 1;

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

%e 7637688, 20930240, 16261120, 3190784, 135168, 1024, 1;

%e 5539372954, 16855075840, 14918594560, 3501457408, 191561728, 2129920, 4096, 1;

%e 13703105571256, 45696508860928, 45120522420224, 12230958252032, 813938245632, 11856248832, 33816576, 16384, 1;

%e 118149647382446899, 427467706869837824, 463647865862488064, 141682892446105600, 11040640699727872, 197960679817216, 745898246144, 538968064, 65536, 1;

%e ...

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

%e Matrix square, T^2, begins:

%e 1;

%e 2, 1;

%e 10, 8, 1;

%e 134, 144, 32, 1;

%e 5296, 7008, 2176, 128, 1;

%e 654070, 1016320, 398848, 33792, 512, 1; ...

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

%e Matrix 4th power, T^4, begins:

%e 1;

%e 4, 1;

%e 36, 16, 1;

%e 876, 544, 64, 1;

%e 63520, 49856, 8448, 256, 1;

%e 14568940, 13677568, 2993152, 133120, 1024, 1; ...

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

%e Matrix 8th power, T^8, begins:

%e 1;

%e 8, 1;

%e 136, 32, 1;

%e 6232, 2112, 128, 1;

%e 854848, 374144, 33280, 512, 1;

%e 373259224, 198715392, 23156736, 528384, 2048, 1; ...

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

%e ...

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

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

%e 1;

%e 1/2, 1;

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

%e 9/2, 12, 8, 1;

%e 58, 184, 160, 32, 1;

%e 4475/2, 8192, 8576, 2304, 128, 1;

%e 269828, 1118048, 1355776, 448512, 34816, 512, 1; ...

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

%e 1;

%e 1/4, 1;

%e 3/8, 1, 1;

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

%e 15/2, 36, 48, 16, 1;

%e 667/4, 928, 1472, 640, 64, 1;

%e 11180, 71600, 131072, 68608, 9216, 256, 1; ...

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

%e 1;

%e 1/8, 1;

%e 5/32, 1/2, 1;

%e 1/4, 3/2, 2, 1;

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

%e 107/8, 120, 288, 192, 32, 1;

%e 977/2, 5336, 14848, 11776, 2560, 128, 1; ...

%e Matrix 16th root, T^(1/8), begins:

%e 1;

%e 1/16, 1;

%e 9/128, 1/4, 1;

%e 9/128, 5/8, 1, 1;

%e 11/128, 2, 6, 4, 1;

%e 1, 16, 64, 64, 16, 1; <== row 5: [T^(1/2^4)](5,k) = (2^k)^(5-k), k=0..5

%e 139/8, 428, 1920, 2304, 768, 64, 1; ...

%o (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]}

%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

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

%Y Cf. A134484.

%Y Cf. A274477 (matrix log).

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Oct 04 2007, Oct 28 2007