W. Lang, Oct 16 2009 A164660/A164661 rationals are the row sums of the rational triangle A164658/A164659: n\m 0 1 2 3 4 5 6 7 8 9 10 ... 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1/2 0 0 0 0 0 0 0 0 0 2 -1 0 2/3 0 0 0 0 0 0 0 0 3 0 -3/2 0 1 0 0 0 0 0 0 0 4 1 0 -8/3 0 8/5 0 0 0 0 0 0 5 0 5/2 0 -5 0 8/3 0 0 0 0 0 6 -1 0 6 0 -48/5 0 32/7 0 0 0 0 7 0 -7/2 0 14 0 56/3 0 8 0 0 0 8 1 0 -32/3 0 32 0 -256/7 0 128/9 0 0 9 0 9/2 0 -30 0 72 0 -72 0 128/5 0 10 -1 0 50/3 0 -80 0 160 0 -1280/9 0 512/11 . . . Row sums give 164660/A164661: [1, 1/2, -1/3, -1/2, -1/15, 1/6, -1/35, -1/6, -1/63, 1/10, -1/99, -1/10, -1/143, 1/14, -1/195, -1/14, -1/255, 1/18, -1/323, -1/18,...]. Numerator triangle A164658: n\m 0 1 2 3 4 5 6 7 8 9 10... 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 2 -1 0 2 0 0 0 0 0 0 0 0 3 0 -3 0 1 0 0 0 0 0 0 0 4 1 0 -8 0 8 0 0 0 0 0 0 5 0 5 0 -5 0 8 0 0 0 0 0 6 -1 0 6 0 -48 0 32 0 0 0 0 7 0 -7 0 14 0 -56 0 8 0 0 0 8 1 0 -32 0 32 0 -256 0 128 0 0 9 0 9 0 -30 0 72 0 -72 0 128 0 10 -1 0 50 0 -80 0 160 0 -1280 0 512 . . . Row sums: A164662 [1, 1, 1, -2, 1, 8, -11, -41, -127, 107, -639, -1372, -3695, 514, ...]. Denominator triangle A164659: n\m 0 1 2 3 4 5 6 7 8 9 10... 0 1 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 2 1 1 3 0 0 0 0 0 0 0 0 3 1 2 1 1 0 0 0 0 0 0 0 4 1 1 3 1 5 0 0 0 0 0 0 5 1 2 1 1 1 3 0 0 0 0 0 6 1 1 1 1 5 1 7 0 0 0 0 7 1 2 1 1 1 3 1 1 0 0 0 8 1 1 3 1 1 1 7 1 9 0 0 9 1 2 1 1 1 1 1 1 1 5 0 10 1 1 3 1 1 1 1 1 9 1 11 . . . Row sums give A164663:[1, 3, 5, 5, 11, 9, 17, 11, 25, 15, 31,...]. ################################## eof ################################ [1], [0, 1/2], [-1, 0, 2/3], [0, -3/2, 0, 1], [1, 0, -8/3, 0, 8/5], [0, 5/2, 0, -5, 0, 8/3], [-1, 0, 6, 0, -48/5, 0, 32/7], [0, -7/2, 0, 14, 0, -56/3, 0, 8], [1, 0, -32/3, 0, 32, 0, -256/7, 0, 128/9], [0, 9/2, 0, -30, 0, 72, 0, -72, 0, 128/5], [-1, 0, 50/3, 0, -80, 0, 160, 0, -1280/9, 0, 512/11], [0, -11/2, 0, 55, 0, -616/3, 0, 352, 0, -1408/5, 0, 256/3], [1, 0, -24, 0, 168, 0, -512, 0, 768, 0, -6144/11, 0, 2048/13], [0, 13/2, 0, -91, 0, 1456/3, 0, -1248, 0, 1664, 0, -3328/3, 0, 2048/7], [-1, 0, 98/3, 0, -1568/5, 0, 1344, 0, -8960/3, 0, 3584, 0, -28672/13, 0, 8192/15], [0, -15/2, 0, 140, 0, -1008, 0, 3600, 0, -7040, 0, 7680, 0, -30720/7, 0, 1024], [1, 0, -128/3, 0, 2688/5, 0, -3072, 0, 28160/3, 0, -16384, 0, 16384, 0, -131072/15, 0, 32768/17],