%I #20 Jul 28 2021 14:03:31
%S 2,4,12,36,96,288,836,2412,7000,20404,59256,172236,500776,1455908,
%T 4232288,12305028
%N Number of binary strings of length 2n having exactly 1 factorization as a concatenation of one or more even-length palindromes.
%C Terms are even by symmetry. - _Michael S. Branicky_, Jul 28 2021
%e a(2) = 4, since the only strings of length 4 with unique factorization are {0011, 0110, 1001, 1100}.
%o (Python)
%o from functools import lru_cache
%o from itertools import product
%o def ispal(s): return s == s[::-1]
%o @lru_cache(maxsize=2*10**7)
%o def f(b): # factorizations of binary string b
%o factorizations = int(len(b) >= 2 and ispal(b))
%o for i in range(2, len(b)-1, 2):
%o factorizations += ispal(b[:i]) * f(b[i:])
%o if factorizations >= 2: return 2 # short circuit on condition
%o return factorizations
%o def a(n):
%o return 2*sum(f("0"+"".join(b))==1 for b in product("01", repeat=2*n-1))
%o print([a(n) for n in range(1, 11)]) # _Michael S. Branicky_, Jul 28 2021
%Y Cf. A241210, A241211.
%K nonn,more
%O 1,1
%A _Jeffrey Shallit_, Apr 17 2014
%E a(10)-a(16) from _Giovanni Resta_, Apr 18 2014