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A370874 Number of partitions of 4n whose xor-sum is 2n.

%I A370874 #41 Mar 27 2024 19:09:37
%S A370874 1,2,4,16,16,65,153,411,165,437,931,2317,4802,10595,21565,43211,5014,
%T A370874 10911,22466,44695,83058,156147,286432,516479,595305,1133892,2111273,
%U A370874 3803940,6731760,11653790,19886537,33275225,916662,1593595,2753582,4676617,7866137
%N A370874 Number of partitions of 4n whose xor-sum is 2n.
%H A370874 Alois P. Heinz, <a href="/A370874/b370874.txt">Table of n, a(n) for n = 0..511</a>
%H A370874 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A370874 a(n) = A050314(4n,2n).
%e A370874 a(0) = 1: the empty partition.
%e A370874 a(1) = 2: 211, 31.
%e A370874 a(2) = 4: 41111, 422, 5111, 62.
%e A370874 a(3) = 16: 42111111, 422211, 4311111, 43221, 4332, 5211111, 52221, 531111, 5322, 6111111, 62211, 6321, 633, 711111, 7221, 732.
%e A370874 a(4) = 16: 811111111, 8221111, 82222, 832111, 83311, 844, 91111111, 922111, 93211, 9331, (10)21111, (10)222, (10)3111, (11)2111, (11)311, (12)4.
%p A370874 b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A370874      `if`(i<1 or ilog2(k)>ilog2(i), 0, b(n, i-1, k)+
%p A370874         b(n-i, min(n-i,i), Bits[Xor](i, k))))
%p A370874     end:
%p A370874 a:= n-> b(4*n$2, 2*n):
%p A370874 seq(a(n), n=0..36);
%Y A370874 Cf. A000041, A050314, A058696.
%K A370874 nonn
%O A370874 0,2
%A A370874 _Alois P. Heinz_, Mar 25 2024