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Cf. A008951, A000070, A000097, A000098, A000710.

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a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022

(PARI) a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022

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The A multiplicative version for factorizations is A339846.

A000041 = integer partitions, strict A000009.

A027187 = partitions of even length, strict A067661, ranked by A028260.

Cf. A000984, ~A001105, `A001700, A001791, A008549, ~A028260, `A053738, A097805, A119620, `A120452, `A126869, A182616, `A202736, A236559, A236913, A236914, ~A239829, A304620, `~A344607, `~A344608, `A345196, `A345927, `A345958, `~A345959, A347443, `~A347448.

From Gus Wiseman, Jan 21 2022: (Start)

Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:

1 11 21 22 32 33 43 44 54

111 1111 221 2211 331 2222 441

2111 111111 2221 3311 3222

11111 3211 221111 3321

22111 11111111 4311

211111 22221

1111111 33111

222111

321111

2211111

21111111

111111111

(End)

From Gus Wiseman, Jan 21 2022: (Start)

The a(1) = 1 through a(9) = 12 partitions with at most one odd part:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(21) (22) (32) (42) (43) (44) (54)

(41) (222) (52) (62) (63)

(221) (61) (422) (72)

(322) (2222) (81)

(421) (432)

(2221) (441)

(522)

(621)

(3222)

(4221)

(22221)

(End)

Table[Length[Select[IntegerPartitions[n], Count[#, _?OddQ]<=1&]], {n, 0, 30}] (* Gus Wiseman, Jan 21 2022 *)

Cf. A000041, A000070, A008951, A000097, A000098, A000710.

The case of alternating sum 0 (equality) is A000070.

The version for factorizations is A339846.

These partitions are ranked by A349150, conjugate A349151.

A000041 = integer partitions.

A027187 = partitions of even length, strict A067661.

A027193 = partitions of odd length, ranked by A026424.

A058695 = partitions of odd numbers.

A103919 = partitions by sum and alternating sum (reverse: A344612).

A277103 = partitions with the same number of odd parts as their conjugate.

Cf. A000984, ~A001105, `A001700, A001791, A008549, ~A028260, `A053738, A097805, A119620, `A120452, `A126869, A182616, `A202736, A236559, A236913, A236914, ~A239829, A304620, `~A344607, `~A344608, `A345196, `A345927, `A345958, `~A345959, A347443, `~A347448.

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