This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A237821 #14 May 27 2023 10:33:02
%S A237821 0,0,1,2,4,7,11,16,25,35,48,68,92,123,164,216,282,367,471,604,769,975,
%T A237821 1225,1542,1924,2395,2968,3669,4514,5547,6781,8280,10071,12229,14796,
%U A237821 17881,21537,25902,31066,37206,44443,53021,63098,74995,88946,105350,124533
%N A237821 Number of partitions of n such that 2*(least part) <= greatest part.
%C A237821 By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - _Gus Wiseman_, May 15 2023
%F A237821 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
%e A237821 a(6) = 7 counts these partitions: 51, 42, 411, 321, 3111, 2211, 21111.
%e A237821 From _Gus Wiseman_, May 15 2023: (Start)
%e A237821 The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
%e A237821 (21) (31) (41) (42) (52)
%e A237821 (211) (221) (51) (61)
%e A237821 (311) (321) (331)
%e A237821 (2111) (411) (421)
%e A237821 (2211) (511)
%e A237821 (3111) (2221)
%e A237821 (21111) (3211)
%e A237821 (4111)
%e A237821 (22111)
%e A237821 (31111)
%e A237821 (211111)
%e A237821 The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
%e A237821 (21) (31) (32) (42) (43)
%e A237821 (211) (41) (51) (52)
%e A237821 (311) (321) (61)
%e A237821 (2111) (411) (322)
%e A237821 (2211) (421)
%e A237821 (3111) (511)
%e A237821 (21111) (3211)
%e A237821 (4111)
%e A237821 (22111)
%e A237821 (31111)
%e A237821 (211111)
%e A237821 (End)
%t A237821 z = 60; q[n_] := q[n] = IntegerPartitions[n];
%t A237821 Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}] (* A237820 *)
%t A237821 Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
%t A237821 Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
%t A237821 Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}] (* A053263 *)
%t A237821 Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)
%Y A237821 The complement is counted by A053263, ranks A081306.
%Y A237821 These partitions have ranks A069900.
%Y A237821 The case of equality is A118096.
%Y A237821 For < instead of <= we have A237820, ranks A362982.
%Y A237821 For >= instead of <= we have A237824, ranks A362981.
%Y A237821 The conjugate partitions have ranks A362980.
%Y A237821 A000041 counts integer partitions, strict A000009.
%Y A237821 A325347 counts partitions with integer median, complement A307683.
%Y A237821 Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.
%K A237821 nonn,easy
%O A237821 1,4
%A A237821 _Clark Kimberling_, Feb 16 2014