cp's OEIS Frontend

A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

%I A238495 #21 Jul 15 2023 14:02:33
%S A238495 1,2,3,4,7,9,14,19,27,36,51,66,90,118,156,201,264,336,434,550,700,880,
%T A238495 1112,1385,1733,2149,2666,3283,4049,4956,6072,7398,9009,10922,13237,
%U A238495 15970,19261,23147,27790,33260,39776,47425,56497,67133,79685,94371,111653
%N A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.
%C A238495 Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - _Gus Wiseman_, Jul 10 2023
%F A238495 From _Gus Wiseman_, Jul 11 2023: (Start)
%F A238495 a(n>2) = A000041(n) - A096373(n-2).
%F A238495 a(n>1) = A000041(n-2) + A002865(n+1).
%F A238495 a(n) = A000041(n+1) - A027336(n).
%F A238495 (End)
%e A238495 a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.
%e A238495 From _Gus Wiseman_, Jul 10 2023 (Start)
%e A238495 The a(2) = 1 through a(8) = 14 partitions:
%e A238495   (2)  (3)   (4)   (5)    (6)     (7)     (8)
%e A238495        (21)  (22)  (32)   (33)    (43)    (44)
%e A238495              (31)  (41)   (42)    (52)    (53)
%e A238495                    (221)  (51)    (61)    (62)
%e A238495                           (222)   (322)   (71)
%e A238495                           (321)   (331)   (332)
%e A238495                           (2211)  (421)   (422)
%e A238495                                   (2221)  (431)
%e A238495                                   (3211)  (521)
%e A238495                                           (2222)
%e A238495                                           (3221)
%e A238495                                           (3311)
%e A238495                                           (4211)
%e A238495                                           (22211)
%e A238495 (End)
%t A238495 Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]
%t A238495 Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* _Gus Wiseman_, Jul 10 2023 *)
%Y A238495 Cf. A096373.
%Y A238495 For mean instead of median we have A000065, ranks A057716.
%Y A238495 The complement is counted by A027336, ranks A364056.
%Y A238495 Rows sums of A359893 if we remove the first column.
%Y A238495 These partitions have ranks A364058.
%Y A238495 A000041 counts integer partitions.
%Y A238495 A008284 counts partitions by length, A058398 by mean.
%Y A238495 A025065 counts partitions with low mean 1, ranks A363949.
%Y A238495 A124943 counts partitions by low median, high A124944.
%Y A238495 A241131 counts partitions with low mode 1, ranks A360015.
%Y A238495 Cf. A000070, A000975, A002865, A110618, A237984, A363488.
%K A238495 nonn,easy
%O A238495 1,2
%A A238495 _Clark Kimberling_, Feb 27 2014
%E A238495 Formula corrected by _Gus Wiseman_, Jul 11 2023