A347448 Number of integer partitions of n with alternating product > 1.

0, 0, 1, 2, 3, 5, 8, 12, 17, 25, 35, 49, 66, 90, 120, 161, 209, 275, 355, 460, 585, 750, 946, 1199, 1498, 1881, 2335, 2909, 3583, 4430, 5428, 6666, 8118, 9912, 12013, 14586, 17592, 21252, 25525, 30695, 36711, 43956, 52382, 62469, 74173, 88132, 104303, 123499

Offset: 0

Author

Gus Wiseman, Sep 16 2021

Keywords

• nonn

Comments

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

Examples

			The a(2) = 1 through a(7) = 12 partitions:
  (2)  (3)   (4)    (5)     (6)      (7)
       (21)  (31)   (32)    (42)     (43)
             (211)  (41)    (51)     (52)
                    (311)   (222)    (61)
                    (2111)  (321)    (322)
                            (411)    (421)
                            (3111)   (511)
                            (21111)  (2221)
                                     (3211)
                                     (4111)
                                     (31111)
                                     (211111)
		

Crossrefs

The strict case is A000009, except that a(0) = a(1) = 0.

Allowing any alternating product >= 1 gives A000041, reverse A344607.

Ranked by A028983 (reverse A347465), which has complement A028982.

The complement is counted by A119620, reverse A347443.

The multiplicative version is A339890, weak A347456, reverse A347705.

The even-length case is A344608.

Allowing any integer reverse-alternating product gives A347445.

Allowing any integer alternating product gives A347446.

The reverse version is A347449, also the odd-length case.

A027187 counts partitions of even length.

A027193 counts partitions of odd length.

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

A347461 counts possible alternating products of partitions.

Cf. A000070, A086543, A100824, A236913, A325534, A325535, A339846, A344654, A345196, A347440, A347444, A347462.

Programs

Maple

a:= n-> (p-> p(n)-p(iquo(n, 2)))(combinat[numbpart]):
seq(a(n), n=0..63);  # Alois P. Heinz, Oct 04 2021

Mathematica

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

Formula

a(n) = A000041(n) - A119620(n).