cp's OEIS Frontend

A182616 Number of partitions of 2n that contain odd parts.

%I A182616 #27 Oct 18 2023 10:07:00
%S A182616 0,1,3,8,17,35,66,120,209,355,585,946,1498,2335,3583,5428,8118,12013,
%T A182616 17592,25525,36711,52382,74173,104303,145698,202268,279153,383145,
%U A182616 523105,710655,960863,1293314,1733281,2313377,3075425,4073085,5374806,7067863,9263076
%N A182616 Number of partitions of 2n that contain odd parts.
%C A182616 Bisection (even part) of A086543.
%F A182616 a(n) = A000041(2*n) - A000041(n).
%e A182616 For n=3 the partitions of 2n are
%e A182616 6 ....................... does not contains odd parts
%e A182616 3 + 3 ................... contains odd parts ........... *
%e A182616 4 + 2 ................... does not contains odd parts
%e A182616 2 + 2 + 2 ............... does not contains odd parts
%e A182616 5 + 1 ................... contains odd parts ........... *
%e A182616 3 + 2 + 1 ............... contains odd parts ........... *
%e A182616 4 + 1 + 1 ............... contains odd parts ........... *
%e A182616 2 + 2 + 1 + 1 ........... contains odd parts ........... *
%e A182616 3 + 1 + 1 + 1 ........... contains odd parts ........... *
%e A182616 2 + 1 + 1 + 1 + 1 ....... contains odd parts ........... *
%e A182616 1 + 1 + 1 + 1 + 1 + 1 ... contains odd parts ........... *
%e A182616 There are 8 partitions of 2n that contain odd parts.
%e A182616 Also p(2n)-p(n) = p(6)-p(3) = 11-3 = 8, where p(n) is the number of partitions of n, so a(3)=8.
%e A182616 From _Gus Wiseman_, Oct 18 2023: (Start)
%e A182616 For n > 0, also the number of integer partitions of 2n that do not contain n, ranked by A366321. For example, the a(1) = 1 through a(4) = 17 partitions are:
%e A182616   (2)  (4)     (6)       (8)
%e A182616        (31)    (42)      (53)
%e A182616        (1111)  (51)      (62)
%e A182616                (222)     (71)
%e A182616                (411)     (332)
%e A182616                (2211)    (521)
%e A182616                (21111)   (611)
%e A182616                (111111)  (2222)
%e A182616                          (3221)
%e A182616                          (3311)
%e A182616                          (5111)
%e A182616                          (22211)
%e A182616                          (32111)
%e A182616                          (221111)
%e A182616                          (311111)
%e A182616                          (2111111)
%e A182616                          (11111111)
%e A182616 (End)
%p A182616 with(combinat): a:= n-> numbpart(2*n) -numbpart(n): seq(a(n), n=0..35);
%t A182616 Table[Length[Select[IntegerPartitions[2n],n>0&&FreeQ[#,n]&]],{n,0,15}] (* _Gus Wiseman_, Oct 11 2023 *)
%t A182616 Table[Length[Select[IntegerPartitions[2n],Or@@OddQ/@#&]],{n,0,15}] (* _Gus Wiseman_, Oct 11 2023 *)
%Y A182616 Cf. A304710.
%Y A182616 Bisection of A086543, with ranks A366322.
%Y A182616 The case of all odd parts is A035294, bisection of A000009.
%Y A182616 The strict case is A365828.
%Y A182616 These partitions have ranks A366530.
%Y A182616 A000041 counts integer partitions, strict A000009.
%Y A182616 A006477 counts partitions with at least one odd and even part, ranks A366532.
%Y A182616 A047967 counts partitions with at least one even part, ranks A324929.
%Y A182616 A086543 counts partitions of n not containing n/2, ranks A366319.
%Y A182616 A366527 counts partitions of 2n with an even part, ranks A366529.
%Y A182616 Cf. A001522, A006827, A058695, A078408, A079122, A231429, A365543, A366321.
%K A182616 nonn
%O A182616 0,3
%A A182616 _Omar E. Pol_, Dec 03 2010
%E A182616 Edited by _Alois P. Heinz_, Dec 03 2010