OFFSET
1,4
COMMENTS
By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023
FORMULA
G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - Seiichi Manyama, May 27 2023
EXAMPLE
a(6) = 7 counts these partitions: 51, 42, 411, 321, 3111, 2211, 21111.
From Gus Wiseman, May 15 2023: (Start)
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
(21) (31) (41) (42) (52)
(211) (221) (51) (61)
(311) (321) (331)
(2111) (411) (421)
(2211) (511)
(3111) (2221)
(21111) (3211)
(4111)
(22111)
(31111)
(211111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
(21) (31) (32) (42) (43)
(211) (41) (51) (52)
(311) (321) (61)
(2111) (411) (322)
(2211) (421)
(3111) (511)
(21111) (3211)
(4111)
(22111)
(31111)
(211111)
(End)
MATHEMATICA
z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}] (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}] (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 16 2014
STATUS
approved