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%I #8 Sep 27 2021 07:57:09
%S 1,0,1,1,3,2,6,4,11,8,18,13,33,22,49,38,79,58,122,90,186,139,268,206,
%T 402,304,569,448,817,636,1152,907,1612,1283,2220,1791,3071,2468,4162,
%U 3409,5655,4634,7597,6283,10171,8478,13491,11336,17906,15088,23513,20012
%N Number of even-length integer partitions of n with integer alternating product.
%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%e The a(2) = 1 through a(9) = 8 partitions:
%e (11) (21) (22) (41) (33) (61) (44) (63)
%e (31) (2111) (42) (2221) (62) (81)
%e (1111) (51) (4111) (71) (3321)
%e (2211) (211111) (2222) (4221)
%e (3111) (3221) (6111)
%e (111111) (3311) (222111)
%e (4211) (411111)
%e (5111) (21111111)
%e (221111)
%e (311111)
%e (11111111)
%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]
%Y Allowing any alternating product >= 1 gives A000041, reverse A344607.
%Y Allowing any alternating product gives A027187, odd bisection A236914.
%Y The Heinz numbers of these partitions are given by A028260 /\ A347457.
%Y The reverse and reciprocal versions are both A035363.
%Y The multiplicative version (factorizations) is A347438, reverse A347439.
%Y The odd-length instead of even-length version is A347444.
%Y Allowing any length gives A347446.
%Y A034008 counts even-length compositions, ranked by A053754.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A119620 counts partitions with alternating product 1.
%Y A325534 counts separable partitions, ranked by A335433.
%Y A325535 counts inseparable partitions, ranked by A335448.
%Y Cf. A000070, A067661, A236913, A304620, A339846, A347437, A347441, A347442, A347445, A347448, A347449, A347454, A347462.
%K nonn
%O 0,5
%A _Gus Wiseman_, Sep 17 2021
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