%I A115198
%S A115198 1,0,1,1,0,1,0,1,1,0,1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,1,
%T A115198 1,1,0,0,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,
%U A115198 0,1,1,1,1,1,1,1,0,0,0
%N A115198 Parity of partitions of n, with 1 for even, 0 for odd (!). The definition 
               follows.
%C A115198 The array with 0 and 1 interchanged is A115199.
%C A115198 The partitions appear in the Abramowitz-Stegun (A-St) order (see the 
               reference, pp. 831-2).
%C 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. Because the partity of permutations is linked, via their cycle 
               structure, to the number of even parts of partitions one uses here 
               1 in order to mark the relevant (even) partitions.
%C A115198 The row length sequence of this array is p(n)=A000041(n) (number of partitions).
%H A115198 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A115198 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 
               1972.
%H A115198 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A115198.text">
               First 10 rows.</a>
%F A115198 a(n,m)= 1 if sum(e(n,m,2*j),j=1..floor(n/2)) is even, else 0, 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)=1.
%e A115198 [1];[0,1];[1,0,1];[0,1,1,0,1];[1,0,0,1,1,0,1];...
%e A115198 a(4,4)=0 because it refers to the 4-th partition of n=4 of the
%e A115198 mentioned A-St ordering, namely to (1^2,2^1)=(1,1,2) which has an odd 
               number
%e A115198 (1) of even parts.
%e A115198 a(5,4)=1 because (1^1,2^2)=(1,2,2) has an even number of even parts
%e A115198 (the number of even parts is in fact 2).
%Y A115198 The sequence of row lengths is A046682 (number of cycle types for even 
               permutations).
%Y A115198 Sequence in context: A128174 A096055 A125144 this_sequence A005614 A071036 
               A166946
%Y A115198 Adjacent sequences: A115195 A115196 A115197 this_sequence A115199 A115200 
               A115201
%K A115198 nonn,easy,tabf
%O A115198 0,1
%A A115198 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 23 
               2006