[math-fun] Question about "central" binomial coefficients binomial(2n+1, n+1)

The OEIS sequence for binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!) (<https://oeis.org/A001700>) begins with 1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052, (0 ≤ n ≤ 25), for which binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1. Is this true for all n ≥ 0 ??? —Dan