%I #11 Nov 08 2013 13:45:42
%S 1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0,0,1,1,1,0,0,1,1,1,0,0,1,
%T 1,1,0,0,1,1,1,0,0,0,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0,
%U 0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1
%N a(n) = the number of ways to express n as a sum d1*(k1!-1) + d2*(k2!-1) + ... + dj*(kj!-1), where all k's are distinct and greater than one and each di is in range [1,ki]; the characteristic function of A219650.
%H Antti Karttunen, <a href="/A230412/b230412.txt">Table of n, a(n) for n = 0..10079</a>
%F a(0)=1 and for n>=1, a(n) = [(A219650(A230413(n-1)) + A230405(A230413(n-1))) = n], where [] stands for Iverson bracket.
%e a(0)=1 as the only solution is an empty sum.
%e 1 can be represented as 1*(2!-1), and this is the only solution, thus a(1) = 1.
%e 2 can be represented (also uniquely) as 2*(2!-1) thus a(2) = 1.
%e 3 and 4 cannot be represented as such a sum, thus a(3) = a(4) = 0.
%e 5 can be represented (uniquely) as 1*(3!-1) thus a(5) = 1.
%e 6 can be represented (uniquely) as 1*(3!-1) + 1*(2!-1), thus a(6) = 1.
%e 7 can be represented (uniquely) as 1*(3!-1) + 2*(2!-1), thus a(7) = 1.
%e 17 can be represented (uniquely) as 3*(3!-1) + 2*(2!-1), thus a(17) = 1.
%o (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)
%o (definec (A230412 n) (if (zero? n) 1 (let ((k (A230413 (- n 1)))) (if (= (+ (A219650 k) (A230405 k)) n) 1 0))))
%Y The characteristic function of A219650. A219658 gives the positions of zeros.
%Y Together with A230413 (the partial sums) can be used to compute A230414, A230423 and A230424.
%Y This sequence relates to the factorial base representation (A007623) in a similar way as A079559 relates to the binary system.
%K nonn
%O 0
%A _Antti Karttunen_, Nov 02 2013