A069888 Self-inverse permutation of natural numbers induced by the automorphism DeepReverse_et_RotateHandshakes! acting on the parenthesizations encoded by A014486.

0, 1, 3, 2, 7, 5, 8, 4, 6, 17, 12, 20, 10, 15, 18, 13, 21, 9, 14, 22, 11, 16, 19, 45, 31, 54, 26, 40, 48, 34, 57, 24, 38, 61, 29, 43, 52, 46, 32, 55, 27, 41, 49, 35, 58, 23, 37, 62, 28, 42, 51, 50, 36, 59, 25, 39, 63, 30, 44, 53, 64, 33, 47, 56, 60, 129, 87, 157, 73, 115, 138