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

%I A347448 #11 Oct 04 2021 17:19:10
%S A347448 0,0,1,2,3,5,8,12,17,25,35,49,66,90,120,161,209,275,355,460,585,750,
%T A347448 946,1199,1498,1881,2335,2909,3583,4430,5428,6666,8118,9912,12013,
%U A347448 14586,17592,21252,25525,30695,36711,43956,52382,62469,74173,88132,104303,123499
%N A347448 Number of integer partitions of n with alternating product > 1.
%C A347448 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%F A347448 a(n) = A000041(n) - A119620(n).
%e A347448 The a(2) = 1 through a(7) = 12 partitions:
%e A347448   (2)  (3)   (4)    (5)     (6)      (7)
%e A347448        (21)  (31)   (32)    (42)     (43)
%e A347448              (211)  (41)    (51)     (52)
%e A347448                     (311)   (222)    (61)
%e A347448                     (2111)  (321)    (322)
%e A347448                             (411)    (421)
%e A347448                             (3111)   (511)
%e A347448                             (21111)  (2221)
%e A347448                                      (3211)
%e A347448                                      (4111)
%e A347448                                      (31111)
%e A347448                                      (211111)
%p A347448 a:= n-> (p-> p(n)-p(iquo(n, 2)))(combinat[numbpart]):
%p A347448 seq(a(n), n=0..63);  # _Alois P. Heinz_, Oct 04 2021
%t A347448 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347448 Table[Length[Select[IntegerPartitions[n],altprod[#]>1&]],{n,0,30}]
%Y A347448 The strict case is A000009, except that a(0) = a(1) = 0.
%Y A347448 Allowing any alternating product >= 1 gives A000041, reverse A344607.
%Y A347448 Ranked by A028983 (reverse A347465), which has complement A028982.
%Y A347448 The complement is counted by A119620, reverse A347443.
%Y A347448 The multiplicative version is A339890, weak A347456, reverse A347705.
%Y A347448 The even-length case is A344608.
%Y A347448 Allowing any integer reverse-alternating product gives A347445.
%Y A347448 Allowing any integer alternating product gives A347446.
%Y A347448 The reverse version is A347449, also the odd-length case.
%Y A347448 A027187 counts partitions of even length.
%Y A347448 A027193 counts partitions of odd length.
%Y A347448 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347448 A347461 counts possible alternating products of partitions.
%Y A347448 Cf. A000070, A086543, A100824, A236913, A325534, A325535, A339846, A344654, A345196, A347440, A347444, A347462.
%K A347448 nonn
%O A347448 0,4
%A A347448 _Gus Wiseman_, Sep 16 2021