A365828 - OEIS

%I #7 Sep 21 2023 08:56:31

%S 1,1,2,3,5,8,12,18,27,39,55,78,108,148,201,270,359,475,623,811,1050,

%T 1351,1728,2201,2789,3517,4418,5527,6887,8553,10585,13055,16055,19685,

%U 24065,29343,35685,43287,52387,63253,76200,91605,109897,131575,157231,187539

%N Number of strict integer partitions of 2n not containing n.

%F a(n) = A000009(2n) - A000009(n) + 1.

%e The a(0) = 1 through a(6) = 12 strict partitions:

%e   ()  (2)  (4)    (6)    (8)      (10)       (12)

%e            (3,1)  (4,2)  (5,3)    (6,4)      (7,5)

%e                   (5,1)  (6,2)    (7,3)      (8,4)

%e                          (7,1)    (8,2)      (9,3)

%e                          (5,2,1)  (9,1)      (10,2)

%e                                   (6,3,1)    (11,1)

%e                                   (7,2,1)    (5,4,3)

%e                                   (4,3,2,1)  (7,3,2)

%e                                              (7,4,1)

%e                                              (8,3,1)

%e                                              (9,2,1)

%e                                              (5,4,2,1)

%t Table[Length[Select[IntegerPartitions[2n],UnsameQ@@#&&FreeQ[#,n]&]],{n,0,30}]

%Y The complement is counted by A111133.

%Y For non-strict partitions we have A182616, complement A000041.

%Y A000009 counts strict integer partitions.

%Y A046663 counts partitions with no submultiset summing to k, strict A365663.

%Y A365827 counts strict partitions not of length 2, complement A140106.

%Y Cf. A008967, A035363, A078408, A079122, A231429, A238628, A344415, A365659.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 20 2023