A347444 Number of odd-length integer partitions of n with integer alternating product.

0, 1, 1, 2, 2, 4, 4, 8, 7, 14, 13, 24, 21, 40, 35, 62, 55, 99, 85, 151, 128, 224, 195, 331, 283, 481, 416, 690, 593, 980, 844, 1379, 1189, 1918, 1665, 2643, 2292, 3630, 3161, 4920, 4299, 6659, 5833, 8931, 7851, 11905, 10526, 15805, 13987, 20872, 18560, 27398

Offset: 0

Author

Gus Wiseman, Sep 14 2021

Keywords

• nonn

Comments

We define the alternating product of a sequence (y_1, ... ,y_k) to be the Product_i y_i^((-1)^(i-1)).

The reverse version (integer reverse-alternating product) is the same.

Examples

			The a(1) = 1 through a(9) = 14 partitions:
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)        (9)
            (111)  (211)  (221)    (222)    (322)      (332)      (333)
                          (311)    (411)    (331)      (422)      (441)
                          (11111)  (21111)  (421)      (611)      (522)
                                            (511)      (22211)    (621)
                                            (22111)    (41111)    (711)
                                            (31111)    (2111111)  (22221)
                                            (1111111)             (32211)
                                                                  (33111)
                                                                  (42111)
                                                                  (51111)
                                                                  (2211111)
                                                                  (3111111)
                                                                  (111111111)
		

Crossrefs

The reciprocal version is A035363.

Allowing any alternating product gives A027193.

The multiplicative version (factorizations) is A347441.

Allowing any length gives A347446, reverse A347445.

Allowing any length and alternating product > 1 gives A347448.

Allowing any reverse-alternating product > 1 gives A347449.

Ranked by A347453.

The even-length instead of odd-length version is A347704.

A000041 counts partitions.

A000302 counts odd-length compositions, ranked by A053738.

A025047 counts wiggly compositions.

A026424 lists numbers with odd bigomega.

A027187 counts partitions of even length, strict A067661.

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

A119620 counts partitions with alternating product 1, ranked by A028982.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A339890 counts odd-length factorizations.

A347437 counts factorizations with integer alternating product.

A347461 counts possible alternating products of partitions.

Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.

Programs

Mathematica

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]