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%I #24 Mar 16 2024 15:23:16
%S 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,
%T 80,85,97,100,118,118,137,144,159,168,193,195,220,233,259,268,303,311,
%U 348,367,399,419,469,483,532,560,610,639,704,732,801,841,908,954
%N Number of partitions of n such that 2*(least part) >= greatest part.
%C 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
%H John Tyler Rascoe, <a href="/A237824/b237824.txt">Table of n, a(n) for n = 1..300</a>
%F G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - _John Tyler Rascoe_, Mar 07 2024
%e a(6) = 7 counts these partitions: 6, 42, 33, 222, 2211, 21111, 111111.
%e From _Gus Wiseman_, May 14 2023: (Start)
%e The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (211) (221) (42) (322) (53)
%e (1111) (2111) (222) (2221) (332)
%e (11111) (2211) (22111) (422)
%e (21111) (211111) (2222)
%e (111111) (1111111) (22211)
%e (221111)
%e (2111111)
%e (11111111)
%e The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (221) (51) (61) (62)
%e (11111) (222) (331) (71)
%e (2211) (2221) (332)
%e (111111) (1111111) (2222)
%e (3311)
%e (22211)
%e (11111111)
%e (End)
%t z = 60; q[n_] := q[n] = IntegerPartitions[n];
%t Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}] (* A237820 *)
%t Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
%t Table[Count[q[n], p_ /; 2 Min[p] == Max[p]], {n, z}] (* A118096 *)
%t Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}] (* A053263 *)
%t Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* this sequence *)
%o (PARI)
%o N=60; x='x+O('x^N);
%o gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));
%o Vec(gf) \\ _John Tyler Rascoe_, Mar 07 2024
%Y Cf. A237821, A118096, A053263.
%Y The complement is counted by A237820, ranks A362982.
%Y For modes instead of middles we have A362619, counted by A171979.
%Y These partitions have ranks A362981.
%Y A000041 counts integer partitions, strict A000009.
%Y A325347 counts partitions with integer median, complement A307683.
%Y Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622.
%K nonn
%O 1,2
%A _Clark Kimberling_, Feb 16 2014
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