a(i,j) tabl heads (triangles) for A119935 and A119932 A119935: Numerators of A^3 matrix, with the lower triangular matrix A of reciprocals of triangle A002024(n,m). i\j 1 2 3 4 5 6 7 8 9 ... 1 1 0 0 0 0 0 0 0 0 2 7 1 0 0 0 0 0 0 0 3 85 19 1 0 0 0 0 0 0 4 415 115 37 1 0 0 0 0 0 5 12019 3799 1489 61 1 0 0 0 0 6 13489 4669 2059 919 91 1 0 0 0 7 726301 268921 128431 64171 7669 127 1 0 0 8 3144919 1227199 621139 334699 178669 3565 169 1 0 9 30300391 12335311 6527971 3714811 2134141 47485 24385 217 1 . . . Row sums give [1,8,105,568,17369,21228,1195621,5510360,55084713,...] = A119934(i), i>=1. ################################################################################################################# A119932: Denominators of A^3 matrix, with the lower triangular matrix A of reciprocals of triangle A002024(n,m). i\j 1 2 3 4 5 6 7 8 9 10 .... 1 1 0 0 0 0 0 0 0 0 0 2 8 8 0 0 0 0 0 0 0 0 3 108 108 27 0 0 0 0 0 0 0 4 576 576 576 64 0 0 0 0 0 0 5 18000 18000 18000 2000 125 0 0 0 0 0 6 21600 21600 21600 21600 5400 216 0 0 0 0 7 1234800 1234800 1234800 1234800 308700 12348 343 0 0 0 8 5644800 5644800 5644800 5644800 5644800 225792 25088 512 0 0 9 57153600 57153600 57153600 57153600 57153600 2286144 2286144 46656 729 0 10 63504000 63504000 63504000 63504000 63504000 63504000 63504000 1296000 81000 1000 . . . Row sums give [1, 16, 243, 1792, 56125, 92016, 5260591, 28475392, 290387673,445906000,...] = A119933(i), i>=1. ####################################################################################################################################### Triangle of rationals A119835(i,j)/A119932(i,j): i\j 1 2 3 4 5 6 7 8 .... 1 1 0 0 0 0 0 0 0 2 7/8 1/8 0 0 0 0 0 0 3 85/108 19/108 1/27 0 0 0 0 0 4 415/576 115/576 37/576 1/64 0 0 0 0 5 12019/18000 3799/18000 1489/18000 61/2000 1/125 0 0 0 6 13489/21600 4669/21600 2059/21600 919/21600 91/5400 1/216 0 0 7 726301/1234800 268921/1234800 128431/1234800 64171/1234800 7669/308700 127/12348 1/343 0 8 3144919/5644800 1227199/5644800 621139/5644800 334699/5644800 178669/5644800 3565/225792 169/25088 1/512 . . . Row i=9: [30300391/57153600, 12335311/57153600, 6527971/57153600, 3714811/57153600, 2134141/57153600, 47485/2286144, 24385/2286144, 217/46656, 1/729], Row i=10: [32160403/63504000, 13560283/63504000, 7435423/63504000, 4410583/63504000, 2671153/63504000, 1597129/63504000, 913789/63504000, 9781/1296000, 271/81000, 1/1000]] Row sums give always 1. ############################################################################################################################################# The triangle A027447 (O. Gerard) is: a(i,j) tabl head (triangle) for A027447 i\j 1 2 3 4 5 6 7 8 ... 1 1 0 0 0 0 0 0 0 2 7 1 0 0 0 0 0 0 3 85 19 4 0 0 0 0 0 4 415 115 37 9 0 0 0 0 5 12019 3799 1489 549 144 0 0 0 6 13489 4669 2059 919 364 100 0 0 7 726301 268921 128431 64171 30676 12700 3600 0 8 3144919 1227199 621139 334699 178669 89125 38025 11025 . . . This results from the rational triangle A119835(i,j)/A119932(i,j) after multiplication of row i with the LCM of the denominators of this row. These numbers are A119936(i)=[1,8,108,576,18000,21600,1234800,5644800,57153600,63504000,...], i>=1. They are i.g. n o t the denominators of the first column. E.g.: Row n=14: LCM is 1818030614400 but the denominator of the first (j=1) column is 259718659200. Therefore a multiplication with the denominator of the first row does not produce an nonnegative integer triangle, starting with row i=14. ############################################################## e.o.f.###########################################################################