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 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. 10
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified September 16 00:43 EDT 2024. Contains 375959 sequences. (Running on oeis4.)