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

1, 2, 4, 16, 16, 65, 153, 411, 165, 437, 931, 2317, 4802, 10595, 21565, 43211, 5014, 10911, 22466, 44695, 83058, 156147, 286432, 516479, 595305, 1133892, 2111273, 3803940, 6731760, 11653790, 19886537, 33275225, 916662, 1593595, 2753582, 4676617, 7866137

Offset: 0

Author

Alois P. Heinz, Mar 25 2024

Keywords

• nonn

Examples

			a(0) = 1: the empty partition.
a(1) = 2: 211, 31.
a(2) = 4: 41111, 422, 5111, 62.
a(3) = 16: 42111111, 422211, 4311111, 43221, 4332, 5211111, 52221, 531111, 5322, 6111111, 62211, 6321, 633, 711111, 7221, 732.
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..511
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A000041, A050314, A058696.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
     `if`(i<1 or ilog2(k)>ilog2(i), 0, b(n, i-1, k)+
        b(n-i, min(n-i,i), Bits[Xor](i, k))))
    end:
a:= n-> b(4*n$2, 2*n):
seq(a(n), n=0..36);

Formula

a(n) = A050314(4n,2n).