cp's OEIS Frontend

A347704 Number of even-length integer partitions of n with integer alternating product.

%I A347704 #8 Sep 27 2021 07:57:09
%S A347704 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 A347704 402,304,569,448,817,636,1152,907,1612,1283,2220,1791,3071,2468,4162,
%U A347704 3409,5655,4634,7597,6283,10171,8478,13491,11336,17906,15088,23513,20012
%N A347704 Number of even-length integer partitions of n with integer alternating product.
%C A347704 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%e A347704 The a(2) = 1 through a(9) = 8 partitions:
%e A347704   (11)  (21)  (22)    (41)    (33)      (61)      (44)        (63)
%e A347704               (31)    (2111)  (42)      (2221)    (62)        (81)
%e A347704               (1111)          (51)      (4111)    (71)        (3321)
%e A347704                               (2211)    (211111)  (2222)      (4221)
%e A347704                               (3111)              (3221)      (6111)
%e A347704                               (111111)            (3311)      (222111)
%e A347704                                                   (4211)      (411111)
%e A347704                                                   (5111)      (21111111)
%e A347704                                                   (221111)
%e A347704                                                   (311111)
%e A347704                                                   (11111111)
%t A347704 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347704 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]
%Y A347704 Allowing any alternating product >= 1 gives A000041, reverse A344607.
%Y A347704 Allowing any alternating product gives A027187, odd bisection A236914.
%Y A347704 The Heinz numbers of these partitions are given by A028260 /\ A347457.
%Y A347704 The reverse and reciprocal versions are both A035363.
%Y A347704 The multiplicative version (factorizations) is A347438, reverse A347439.
%Y A347704 The odd-length instead of even-length version is A347444.
%Y A347704 Allowing any length gives A347446.
%Y A347704 A034008 counts even-length compositions, ranked by A053754.
%Y A347704 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347704 A119620 counts partitions with alternating product 1.
%Y A347704 A325534 counts separable partitions, ranked by A335433.
%Y A347704 A325535 counts inseparable partitions, ranked by A335448.
%Y A347704 Cf. A000070, A067661, A236913, A304620, A339846, A347437, A347441, A347442, A347445, A347448, A347449, A347454, A347462.
%K A347704 nonn
%O A347704 0,5
%A A347704 _Gus Wiseman_, Sep 17 2021