Not an answer, but an amusing observation: The determinant of the matrix usually (but not alwasy) has denominator a perfect square (empirically), while the numerator always seems to have a huge prime factor.
n=1: det = 1/1
n=2: det=7/(12^2)
n=3 det = 647/(2160^2)
n=4 det = (19 * 571)/(672000^2)
n=5 det = (179 * 179357)/(7*4233600000^2)
n=6 det = (97 * 157 * 384191938531)/(186313420339200000^2)
n=7 det = (23 * 1280587616051046200369)/(2067909047925770649600000^2)
n=8 det = (317 * 6337 * 25997 * 87403 * 511645991608091)/(365356847125734485878112256000000^2)
(and some more. Note that mysterious 7 which keeps popping up in the denominator, for n=5,9,11 )
n=9 det = (55091 * 7731550926975871647518813143593349)/(7 * 146968826339795671126721851844198400000000^2)
n=10 det = (257 * 47360083 * 530916328215423816923887043836865928533)/(15402297982638230438765209613012092908994560000000000^2)
n=11 det = (31 * 1193 * 2647 * 538580971 * 957346850101 * 71222443011485886519799225151)/(7 * 175251348661711183890804992735665222783492739007774720000000000^2))