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A347445 Number of integer partitions of n with integer reverse-alternating product.

%I A347445 #9 Sep 27 2021 07:55:16
%S A347445 1,1,2,2,4,4,7,8,12,14,20,24,32,40,50,62,77,99,115,151,170,224,251,
%T A347445 331,360,481,517,690,728,980,1020,1379,1420,1918,1962,2643,2677,3630,
%U A347445 3651,4920,4926,6659,6625,8931,8853,11905,11781,15805,15562,20872,20518
%N A347445 Number of integer partitions of n with integer reverse-alternating product.
%C A347445 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.
%e A347445 The a(1) = 1 through a(8) = 12 partitions:
%e A347445   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A347445        (11)  (111)  (22)    (221)    (33)      (322)      (44)
%e A347445                     (211)   (311)    (222)     (331)      (332)
%e A347445                     (1111)  (11111)  (411)     (421)      (422)
%e A347445                                      (2211)    (511)      (611)
%e A347445                                      (21111)   (22111)    (2222)
%e A347445                                      (111111)  (31111)    (3311)
%e A347445                                                (1111111)  (22211)
%e A347445                                                           (41111)
%e A347445                                                           (221111)
%e A347445                                                           (2111111)
%e A347445                                                           (11111111)
%t A347445 revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347445 Table[Length[Select[IntegerPartitions[n],IntegerQ[revaltprod[#]]&]],{n,0,30}]
%Y A347445 Allowing any reverse-alternating product >= 1 gives A344607.
%Y A347445 Allowing any reverse-alternating product < 1 gives A344608.
%Y A347445 The multiplicative version is A347442, unreversed A347437.
%Y A347445 Allowing any reverse-alternating product <= 1 gives A347443.
%Y A347445 Restricting to odd length gives A347444, ranked by A347453.
%Y A347445 The unreversed version is A347446, ranked by A347457.
%Y A347445 Allowing any reverse-alternating product > 1 gives A347449.
%Y A347445 Ranked by A347454.
%Y A347445 A000041 counts partitions, with multiplicative version A001055.
%Y A347445 A027187 counts partitions of even length.
%Y A347445 A027193 counts partitions of odd length.
%Y A347445 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347445 A325534 counts separable partitions, ranked by A335433.
%Y A347445 A325535 counts inseparable partitions, ranked by A335448.
%Y A347445 A339890 counts factorizations with alternating product > 1, reverse A347705.
%Y A347445 A347462 counts possible reverse-alternating products of partitions.
%Y A347445 Cf. A025047, A067661, A119620, A344654, A344740, A347439, A347440, A347448, A347450, A347451, A347461, A347463, A347704.
%K A347445 nonn
%O A347445 0,3
%A A347445 _Gus Wiseman_, Sep 14 2021