%I #4 Mar 14 2015 13:11:04
%S 0,1,0,0,1,0,1,0,0,1,0,0,1,1,0,0,1,0,1,0,0,0,1,1,1,0,0,1,0,0,1,1,1,0,
%T 0,0,0,1,1,1,0,0,1,0,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,1,0,0,1,
%U 1,1,1,0,0,0,0,0,0,0,1,1,1
%N Parity of partitions of n, with 0 for even, 1 for odd. The definition follows.
%C The main array with 0 and 1 interchanged is A115198.
%C A partition of n is (here) called even, resp. odd, if the number of even parts is even, resp. odd. A partition with no (0) even part is therefore even.
%C The row length sequence of this triangle is p(n)=A000041(n) (number of partitions).
%C See the W. Lang link under A115198 for the first 10 rows where 0 and 1 should be swapped for this a(n,m) entry.
%F a(n,m)= 0 if sum(e(n,m,2*j),j=1..floor(n/2)) is even, else 1, with the exponents e(n,m,k) of the m-th partition of n in the A-St order; i.e. the sum of the exponents of the even parts of the partition (1^e(n,m,1),2^e(n,m,2),..., n^e(n,m,n)) is even iff a(n,m)=0.
%e [0];[1,0];[0,1,0];[1,0,0,1,0];[0,1,1,0,0,1,0];...
%e a(5,4)=0 because the 4th partition of n=5, (1^1,2^2)=(1,2,2), in the A-St order, has an even number of even parts (the number of even parts is in fact 2).
%K nonn,easy,tabf
%O 0,1
%A _Wolfdieter Lang_, Feb 23 2006
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