A361861 Number of integer partitions of n where the median is twice the minimum.

0, 0, 0, 1, 1, 1, 2, 5, 5, 8, 11, 16, 20, 28, 38, 53, 67, 87, 111, 146, 183, 236, 297, 379, 471, 591, 729, 909, 1116, 1376, 1682, 2065, 2507, 3055, 3699, 4482, 5395, 6501, 7790, 9345, 11153, 13316, 15839, 18844, 22333, 26466, 31266, 36924, 43478, 51177

Offset: 1

Author

Gus Wiseman, Apr 02 2023

Keywords

• nonn

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(4) = 1 through a(11) = 11 partitions:
  (31)  (221)  (321)  (421)   (62)     (621)    (442)     (542)
                      (2221)  (521)    (4221)   (721)     (821)
                              (3221)   (4311)   (5221)    (6221)
                              (3311)   (22221)  (5311)    (6311)
                              (22211)  (32211)  (32221)   (33221)
                                                (33211)   (42221)
                                                (42211)   (43211)
                                                (222211)  (52211)
                                                          (222221)
                                                          (322211)
                                                          (2222111)
The partition (3,2,2,2,1,1) has median 2 and minimum 1, so is counted under a(11).
The partition (5,4,2) has median 4 and minimum 2, so is counted under a(11).
		

Crossrefs

For maximum instead of median we have A118096.

For length instead of median we have A237757, without the coefficient A006141.

With minimum instead of twice minimum we have A361860.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, A058398 by mean.

A325347 counts partitions with integer median, complement A307683.

A359893 and A359901 count partitions by median, odd-length A359902.

A360005 gives twice median of prime indices, distinct A360457.

Cf. A027193, A039900, A053263, A067659, A111907, A116608, A237753, A237755, A237824, A361848, A361853.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],2*Min@@#==Median[#]&]],{n,30}]