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 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 (list; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=1..59. EXAMPLE a(6) = 7 counts these partitions: 6, 42, 33, 222, 2211, 21111, 111111. From Gus Wiseman, May 14 2023: (Start) 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 Cf. A237821, A118096, A053263. The complement is counted by A237820, ranks A362982. For modes instead of middles we have A362619, counted by A171979. These partitions have ranks A362981. A000041 counts integer partitions, strict A000009. A325347 counts partitions with integer median, complement A307683. Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622. Sequence in context: A338671 A343246 A348531 * A227972 A266620 A222415 Adjacent sequences: A237821 A237822 A237823 * A237825 A237826 A237827 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 16 2014 STATUS approved

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Last modified June 8 14:02 EDT 2023. Contains 363165 sequences. (Running on oeis4.)