cp's OEIS Frontend

A347444 Number of odd-length integer partitions of n with integer alternating product.

%I A347444 #13 Oct 27 2021 09:45:31
%S A347444 0,1,1,2,2,4,4,8,7,14,13,24,21,40,35,62,55,99,85,151,128,224,195,331,
%T A347444 283,481,416,690,593,980,844,1379,1189,1918,1665,2643,2292,3630,3161,
%U A347444 4920,4299,6659,5833,8931,7851,11905,10526,15805,13987,20872,18560,27398
%N A347444 Number of odd-length integer partitions of n with integer alternating product.
%C A347444 We define the alternating product of a sequence (y_1, ... ,y_k) to be the Product_i y_i^((-1)^(i-1)).
%C A347444 The reverse version (integer reverse-alternating product) is the same.
%e A347444 The a(1) = 1 through a(9) = 14 partitions:
%e A347444   (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)        (9)
%e A347444             (111)  (211)  (221)    (222)    (322)      (332)      (333)
%e A347444                           (311)    (411)    (331)      (422)      (441)
%e A347444                           (11111)  (21111)  (421)      (611)      (522)
%e A347444                                             (511)      (22211)    (621)
%e A347444                                             (22111)    (41111)    (711)
%e A347444                                             (31111)    (2111111)  (22221)
%e A347444                                             (1111111)             (32211)
%e A347444                                                                   (33111)
%e A347444                                                                   (42111)
%e A347444                                                                   (51111)
%e A347444                                                                   (2211111)
%e A347444                                                                   (3111111)
%e A347444                                                                   (111111111)
%t A347444 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347444 Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]
%Y A347444 The reciprocal version is A035363.
%Y A347444 Allowing any alternating product gives A027193.
%Y A347444 The multiplicative version (factorizations) is A347441.
%Y A347444 Allowing any length gives A347446, reverse A347445.
%Y A347444 Allowing any length and alternating product > 1 gives A347448.
%Y A347444 Allowing any reverse-alternating product > 1 gives A347449.
%Y A347444 Ranked by A347453.
%Y A347444 The even-length instead of odd-length version is A347704.
%Y A347444 A000041 counts partitions.
%Y A347444 A000302 counts odd-length compositions, ranked by A053738.
%Y A347444 A025047 counts wiggly compositions.
%Y A347444 A026424 lists numbers with odd bigomega.
%Y A347444 A027187 counts partitions of even length, strict A067661.
%Y A347444 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347444 A119620 counts partitions with alternating product 1, ranked by A028982.
%Y A347444 A325534 counts separable partitions, ranked by A335433.
%Y A347444 A325535 counts inseparable partitions, ranked by A335448.
%Y A347444 A339890 counts odd-length factorizations.
%Y A347444 A347437 counts factorizations with integer alternating product.
%Y A347444 A347461 counts possible alternating products of partitions.
%Y A347444 Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.
%K A347444 nonn
%O A347444 0,4
%A A347444 _Gus Wiseman_, Sep 14 2021