

A237824


Number of partitions of n such that 2*(least part) >= greatest part.


29



1, 2, 3, 4, 5, 7, 7, 10, 11, 13, 14, 19, 18, 23, 25, 29, 30, 38, 37, 46, 48, 54, 57, 70, 69, 80, 85, 97, 100, 118, 118, 137, 144, 159, 168, 193, 195, 220, 233, 259, 268, 303, 311, 348, 367, 399, 419, 469, 483, 532, 560, 610, 639, 704, 732, 801, 841, 908, 954
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OFFSET

1,2


COMMENTS

By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622.  Gus Wiseman, May 14 2023


LINKS



EXAMPLE

a(6) = 7 counts these partitions: 6, 42, 33, 222, 2211, 21111, 111111.
The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (211) (221) (42) (322) (53)
(1111) (2111) (222) (2221) (332)
(11111) (2211) (22111) (422)
(21111) (211111) (2222)
(111111) (1111111) (22211)
(221111)
(2111111)
(11111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(2211) (2221) (332)
(111111) (1111111) (2222)
(3311)
(22211)
(11111111)
(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

These partitions have ranks A362981.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



