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