A325269 - OEIS

%I #17 Feb 02 2023 05:08:10

%S 0,0,0,2,3,6,9,14,20,29,40,55,75,100,133,175,229,296,383,489,625,791,

%T 1000,1254,1573,1957,2434,3009,3716,4564,5602,6841,8347,10142,12308,

%U 14882,17975,21636,26013,31184,37336,44582,53172,63260,75173,89133,105556

%N Number of integer partitions of n with 2 distinct parts or at least 3 parts.

%C The Heinz numbers of these partitions are given by A080257.

%C Partitions with 2 distinct parts are in A002133(n). Partitions with at least 3 parts are in A004250(n). Some partitions are in both subsets, so A002133(n)+A004250(n) >= a(n). - _R. J. Mathar_, Dec 13 2022

%F conjecture: a(n) = A000041(n) - A000034(n-1), n>0. - _R. J. Mathar_, Dec 13 2022

%e The a(1) = 1 through a(8) = 20 partitions:

%e   (21)   (31)    (32)     (42)      (43)       (53)

%e   (111)  (211)   (41)     (51)      (52)       (62)

%e          (1111)  (221)    (222)     (61)       (71)

%e                  (311)    (321)     (322)      (332)

%e                  (2111)   (411)     (331)      (422)

%e                  (11111)  (2211)    (421)      (431)

%e                           (3111)    (511)      (521)

%e                           (21111)   (2221)     (611)

%e                           (111111)  (3211)     (2222)

%e                                     (4111)     (3221)

%e                                     (22111)    (3311)

%e                                     (31111)    (4211)

%e                                     (211111)   (5111)

%e                                     (1111111)  (22211)

%e                                                (32111)

%e                                                (41111)

%e                                                (221111)

%e                                                (311111)

%e                                                (2111111)

%e                                                (11111111)

%p A325269 := proc(n)

%p     local a,p,s ;

%p     a := 0 ;

%p     for p in combinat[partition](n) do

%p         s := convert(p,set) ;

%p         if nops(p) >= 3 or nops(s) = 2 then

%p             a := a+1 ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A325269(n),n=0..40) ; # _R. J. Mathar_, Dec 13 2022

%t Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2||Length[#]>2&]],{n,0,30}]

%Y Cf. A001221, A001222, A001358, A001399, A007774, A008284, A060687, A080257, A090858, A116608, A325244.

%Y Cf. A000041, A002133, A004250.

%K nonn

%O 0,4

%A _Gus Wiseman_, Apr 18 2019