a(i,j) tabl head (triangle) for A119947 (numerators of A^2 of matrix A with elements a(i,j) = 1/A002024(i,j)) i\j 1 2 3 4 5 6 7 8 9 10... 1 1 0 0 0 0 0 0 0 0 0 2 3 1 0 0 0 0 0 0 0 0 3 11 5 1 0 0 0 0 0 0 0 4 25 13 7 1 0 0 0 0 0 0 5 137 77 47 9 1 0 0 0 0 0 6 49 29 19 37 11 1 0 0 0 0 7 363 223 153 319 107 13 1 0 0 0 8 761 481 341 743 533 73 15 1 0 0 9 7129 4609 3349 2509 1879 275 191 17 1 0 10 7381 4861 3601 2761 2131 1627 1207 121 19 1 . . . The row sums give: [1,4,17,46,271,146,1179,2948,19959,23710,...]= A119949(n), n>=1. The corresponding denominator triangle A119948 is: (denominators of A^2 of matrix A with elements a(i,j) = 1/A002024(i,j)) a(i,j) tabl head (triangle) for A119948 i\j 1 2 3 4 5 6 7 8 9 10 ... 1 1 0 0 0 0 0 0 0 0 0 2 4 4 0 0 0 0 0 0 0 0 3 18 18 9 0 0 0 0 0 0 0 4 48 48 48 16 0 0 0 0 0 0 5 300 300 300 100 25 0 0 0 0 0 6 120 120 120 360 180 36 0 0 0 0 7 980 980 980 2940 1470 294 49 0 0 0 8 2240 2240 2240 6720 6720 1344 448 64 0 0 9 22680 22680 22680 22680 22680 4536 4536 648 81 0 10 25200 25200 25200 25200 25200 25200 25200 3600 900 100 . . . The row sums give: [1,8,45,160,1025,936,7693,22016,123201,181000,...]= A119950(n), n>=1. ################################################################################################################# The triangle of rationals A119947(i,j)/ A119948(i,j) is i\j 1 2 3 4 5 6 7 8 9 10 ... 1 1 0 0 0 0 0 0 0 0 0 2 3/4 1/4 0 0 0 0 0 0 0 0 3 11/18 5/18 1/9 0 0 0 0 0 0 0 4 25/48 13/48 7/48 1/16 0 0 0 0 0 0 5 137/300 77/300 47/300 9/100 1/25 0 0 0 0 0 6 49/120 29/120 19/120 37/360 11/180 1/36 0 0 0 0 7 363/980 223/980 153/980 319/2940 107/1470 13/294 1/49 0 0 0 8 761/2240 481/2240 341/2240 743/6720 533/6720 73/1344 15/448 1/64 0 0 9 7129/22680 4609/22680 3349/22680 2509/22680 1879/22680 275/4536 191/4536 17/648 1/81 0 10 7381/25200 4861/25200 3601/25200 2761/25200 2131/25200 1627/25200 1207/25200 121/3600 19/900 1/100 . . . The least common multiple (LCM) of the denominators of row n, n>=1, is [1, 4, 18, 48, 300, 360, 2940, 6720, 22680, 25200, ...] which coincides with A081528(n), n>=1. The rows add always to 1. ################################################################################################################### The triangle of nonnegative numbers obtained from the rational triangle A119947(i,j)/ A119948(i,j) by multiplying each row with the LCM of its denominators is A027446: a(i,j) tabl head (triangle) for A027446 i\j 1 2 3 4 5 6 7 8 9 10 ... 1 1 0 0 0 0 0 0 0 0 0 2 3 1 0 0 0 0 0 0 0 0 3 11 5 2 0 0 0 0 0 0 0 4 25 13 7 3 0 0 0 0 0 0 5 137 77 47 27 12 0 0 0 0 0 6 147 87 57 37 22 10 0 0 0 0 7 1089 669 459 319 214 130 60 0 0 0 8 2283 1443 1023 743 533 365 225 105 0 0 9 7129 4609 3349 2509 1879 1375 955 595 280 0 10 7381 4861 3601 2761 2131 1627 1207 847 532 252 . . . Row sums give [1, 4, 18, 48, 300, 360, 2940, 6720, 22680, 25200,...] which coincides with A081528(i), i>=1. A081528(n):=n*LCM{1,2,...,n} appeared also as LCM of the denominators of row i. The analogue of this triangle for the matrix A^3 instead of A^2 is A027447. In this cubic case the rational triangle is A119935(i,j)/A119932(i,j). #################################################### e.o.f. ###########################################################