Asmany Number Verification

Asmany strings are strings of '0's and '1's that have as many 00 as 11. A string such as 00110001 consists of 3 "00" and 1 "11". Of course this is not an Asmany string. 0011, 1100, 000111000111 are Asmany strings. An L'th Asmany number is the number of Asmany strings of length L for all positive integers L. For esoteric purposes Chef had an oracle (a device) that was capable of answering whether