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 A237824 #43 Jun 16 2025 09:43:16
%S A237824 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 A237824 80,85,97,100,118,118,137,144,159,168,193,195,220,233,259,268,303,311,
%U A237824 348,367,399,419,469,483,532,560,610,639,704,732,801,841,908,954
%N A237824 Number of partitions of n such that 2*(least part) >= greatest part.
%C A237824 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 A237824 Vaclav Kotesovec, <a href="/A237824/b237824.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..300 from John Tyler Rascoe)
%F A237824 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
%F A237824 G.f.: Sum_{k>=1} x^k / Product_{j=k..2*k} (1 - x^j). - _Vaclav Kotesovec_, Jun 13 2025
%F A237824 a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4) * sqrt(2*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 14 2025
%e A237824 a(6) = 7 counts these partitions: 6, 42, 33, 222, 2211, 21111, 111111.
%e A237824 From _Gus Wiseman_, May 14 2023: (Start)
%e A237824 The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
%e A237824 (1) (2) (3) (4) (5) (6) (7) (8)
%e A237824 (11) (21) (22) (32) (33) (43) (44)
%e A237824 (111) (211) (221) (42) (322) (53)
%e A237824 (1111) (2111) (222) (2221) (332)
%e A237824 (11111) (2211) (22111) (422)
%e A237824 (21111) (211111) (2222)
%e A237824 (111111) (1111111) (22211)
%e A237824 (221111)
%e A237824 (2111111)
%e A237824 (11111111)
%e A237824 The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
%e A237824 (1) (2) (3) (4) (5) (6) (7) (8)
%e A237824 (11) (21) (22) (32) (33) (43) (44)
%e A237824 (111) (31) (41) (42) (52) (53)
%e A237824 (1111) (221) (51) (61) (62)
%e A237824 (11111) (222) (331) (71)
%e A237824 (2211) (2221) (332)
%e A237824 (111111) (1111111) (2222)
%e A237824 (3311)
%e A237824 (22211)
%e A237824 (11111111)
%e A237824 (End)
%t A237824 z = 60; q[n_] := q[n] = IntegerPartitions[n];
%t A237824 Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}] (* A237820 *)
%t A237824 Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
%t A237824 Table[Count[q[n], p_ /; 2 Min[p] == Max[p]], {n, z}] (* A118096 *)
%t A237824 Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}] (* A053263 *)
%t A237824 Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* this sequence *)
%t A237824 (* or *)
%t A237824 nmax = 100; Rest[CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j,k,2*k}], {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 13 2025 *)
%t A237824 (* or *)
%t A237824 nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax+1)]; s += x^k/(1 - x^k)/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 14 2025 *)
%o A237824 (PARI)
%o A237824 N=60; x='x+O('x^N);
%o A237824 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 A237824 Vec(gf) \\ _John Tyler Rascoe_, Mar 07 2024
%Y A237824 Cf. A237821, A118096, A053263.
%Y A237824 The complement is counted by A237820, ranks A362982.
%Y A237824 For modes instead of middles we have A362619, counted by A171979.
%Y A237824 These partitions have ranks A362981.
%Y A237824 A000041 counts integer partitions, strict A000009.
%Y A237824 A325347 counts partitions with integer median, complement A307683.
%Y A237824 Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622, A384426.
%K A237824 nonn
%O A237824 1,2
%A A237824 _Clark Kimberling_, Feb 16 2014