cp's OEIS Frontend

A347446 Number of integer partitions of n with integer alternating product.

%I A347446 #9 Sep 27 2021 07:55:26
%S A347446 1,1,2,3,5,6,10,12,18,22,31,37,54,62,84,100,134,157,207,241,314,363,
%T A347446 463,537,685,785,985,1138,1410,1616,1996,2286,2801,3201,3885,4434,
%U A347446 5363,6098,7323,8329,9954,11293,13430,15214,18022,20383,24017,27141,31893,35960
%N A347446 Number of integer partitions of n with integer alternating product.
%C A347446 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%e A347446 The a(1) = 1 through a(7) = 12 partitions:
%e A347446   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A347446        (11)  (21)   (22)    (41)     (33)      (61)
%e A347446              (111)  (31)    (221)    (42)      (322)
%e A347446                     (211)   (311)    (51)      (331)
%e A347446                     (1111)  (2111)   (222)     (421)
%e A347446                             (11111)  (411)     (511)
%e A347446                                      (2211)    (2221)
%e A347446                                      (3111)    (4111)
%e A347446                                      (21111)   (22111)
%e A347446                                      (111111)  (31111)
%e A347446                                                (211111)
%e A347446                                                (1111111)
%t A347446 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347446 Table[Length[Select[IntegerPartitions[n],IntegerQ[altprod[#]]&]],{n,0,30}]
%Y A347446 Allowing any reverse-alternating product >= 1 gives A344607.
%Y A347446 Allowing any alternating product <= 1 gives A119620, reverse A347443.
%Y A347446 Allowing any reverse-alternating product < 1 gives A344608.
%Y A347446 The multiplicative version (factorizations) is A347437, reverse A347442.
%Y A347446 The odd-length case is A347444, ranked by A347453.
%Y A347446 The reverse version is A347445, ranked by A347454.
%Y A347446 Allowing any alternating product > 1 gives A347448, reverse A347449.
%Y A347446 Ranked by A347457.
%Y A347446 The even-length case is A347704.
%Y A347446 A000041 counts partitions.
%Y A347446 A027187 counts partitions of even length.
%Y A347446 A027193 counts partitions of odd length.
%Y A347446 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347446 A325534 counts separable partitions, ranked by A335433.
%Y A347446 A325535 counts inseparable partitions, ranked by A335448.
%Y A347446 A347461 counts possible alternating products of partitions.
%Y A347446 Cf. A025047, A067661, A339890, A347450, A344654, A344740, A347439, A347440, A347451, A347463, A347705.
%K A347446 nonn
%O A347446 0,3
%A A347446 _Gus Wiseman_, Sep 15 2021