A115199 Parity of partitions of n, with 0 for even, 1 for odd. The definition follows.

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, 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, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1

Offset: 0

Author

Wolfdieter Lang, Feb 23 2006

Keywords

• nonn
• easy
• tabf

Comments

The main array with 0 and 1 interchanged is A115198.

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.

The row length sequence of this triangle is p(n)=A000041(n) (number of partitions).

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.

Examples

			[0];[1,0];[0,1,0];[1,0,0,1,0];[0,1,1,0,0,1,0];...
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).
		

Formula

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.