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

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..511

Wikipedia, Partition (number theory)

Formula

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

Example

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.

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);

Crossrefs

Cf. A000041, A050314, A058696.

Sequence in context: A155893 A196202 A135569 * A337109 A210579 A217863

Adjacent sequences: A370871 A370872 A370873 * A370875 A370876 A370877

Keyword

nonn

Author

Alois P. Heinz, Mar 25 2024

Status

approved