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Number of partitions of 2n into distinct parts whose bitwise XOR equals 0.
2

%I #11 Apr 12 2019 08:59:27

%S 1,0,0,1,0,1,1,8,1,2,1,9,2,9,10,80,38,39,39,47,40,48,91,126,85,86,136,

%T 165,174,244,512,1187,1117,1135,1741,1374,1932,1990,4284,2665,3200,

%U 2832,5566,3904,6182,6676,15426,11394,12304,11223,15799,12630,15956,17969

%N Number of partitions of 2n into distinct parts whose bitwise XOR equals 0.

%C There are no partitions of 2n+1 into distinct parts whose bitwise XOR equals 0.

%H Alois P. Heinz, <a href="/A307506/b307506.txt">Table of n, a(n) for n = 0..750</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bitwise operation">Bitwise operation</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%F a(n) = A307505(2n,0).

%p b:= proc(n, i, k) option remember; `if`(i*(i+2)/2<n, 0,

%p `if`(n=0, `if`(k=0, 1, 0), b(n, i-1, k)+

%p b(n-i, min(n-i, i-1), Bits[Xor](i, k))))

%p end:

%p a:= n-> b(2*n$2, 0):

%p seq(a(n), n=0..60);

%Y Bisection (even part) of column k=0 of A307505.

%K nonn,base

%O 0,8

%A _Alois P. Heinz_, Apr 11 2019