OEIS A164660, final FORMULA (29 May 2016): Chebyshev numerators and floor products

We prove Anthony Browne's conjecture in OEIS A164660, final FORMULA (29 May 2016): for every positive integer n, the reduced numerator of the integral from 0 to 1 of T_(n+1)(x) equals the stated floor product and (-1)^(n/rad(n)). Both sides are +1 exactly when 4 divides n. The same calculation proves the preceding rational-integral conjecture and gives its denominator sequence A164661. More generally, replacing the exponent n in the product by any integer r >= 2, and translating its interval by any integer, leaves the product unchanged. We give the exact root count for every r >= 1 and correct two coefficient-table discrepancies in the linked sources.