A347444 - OEIS

%I #13 Oct 27 2021 09:45:31

%S 0,1,1,2,2,4,4,8,7,14,13,24,21,40,35,62,55,99,85,151,128,224,195,331,

%T 283,481,416,690,593,980,844,1379,1189,1918,1665,2643,2292,3630,3161,

%U 4920,4299,6659,5833,8931,7851,11905,10526,15805,13987,20872,18560,27398

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

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

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

%e The a(1) = 1 through a(9) = 14 partitions:

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

%e             (111)  (211)  (221)    (222)    (322)      (332)      (333)

%e                           (311)    (411)    (331)      (422)      (441)

%e                           (11111)  (21111)  (421)      (611)      (522)

%e                                             (511)      (22211)    (621)

%e                                             (22111)    (41111)    (711)

%e                                             (31111)    (2111111)  (22221)

%e                                             (1111111)             (32211)

%e                                                                   (33111)

%e                                                                   (42111)

%e                                                                   (51111)

%e                                                                   (2211111)

%e                                                                   (3111111)

%e                                                                   (111111111)

%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];

%t Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

%Y The reciprocal version is A035363.

%Y Allowing any alternating product gives A027193.

%Y The multiplicative version (factorizations) is A347441.

%Y Allowing any length gives A347446, reverse A347445.

%Y Allowing any length and alternating product > 1 gives A347448.

%Y Allowing any reverse-alternating product > 1 gives A347449.

%Y Ranked by A347453.

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

%Y A000041 counts partitions.

%Y A000302 counts odd-length compositions, ranked by A053738.

%Y A025047 counts wiggly compositions.

%Y A026424 lists numbers with odd bigomega.

%Y A027187 counts partitions of even length, strict A067661.

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

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

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A339890 counts odd-length factorizations.

%Y A347437 counts factorizations with integer alternating product.

%Y A347461 counts possible alternating products of partitions.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 14 2021