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 A086543 #23 Oct 15 2023 09:30:41
%S A086543 0,1,1,3,3,7,8,15,17,30,35,56,66,101,120,176,209,297,355,490,585,792,
%T A086543 946,1255,1498,1958,2335,3010,3583,4565,5428,6842,8118,10143,12013,
%U A086543 14883,17592,21637,25525,31185,36711,44583,52382,63261,74173,89134,104303,124754,145698,173525,202268
%N A086543 Number of partitions of n with at least one odd part.
%C A086543 From _Gus Wiseman_, Oct 12 2023: (Start)
%C A086543 Also the number of integer partitions of n whose greatest part is not n/2, ranked by A366319. The a(1) = 1 through a(7) = 15 partitions are:
%C A086543 (1) (2) (3) (4) (5) (6) (7)
%C A086543 (21) (31) (32) (42) (43)
%C A086543 (111) (1111) (41) (51) (52)
%C A086543 (221) (222) (61)
%C A086543 (311) (411) (322)
%C A086543 (2111) (2211) (331)
%C A086543 (11111) (21111) (421)
%C A086543 (111111) (511)
%C A086543 (2221)
%C A086543 (3211)
%C A086543 (4111)
%C A086543 (22111)
%C A086543 (31111)
%C A086543 (211111)
%C A086543 (1111111)
%C A086543 Compare to the a(1) = 1 through a(7) = 15 partitions with at least one odd part, ranked by A366322:
%C A086543 (1) (11) (3) (31) (5) (33) (7)
%C A086543 (21) (211) (32) (51) (43)
%C A086543 (111) (1111) (41) (321) (52)
%C A086543 (221) (411) (61)
%C A086543 (311) (2211) (322)
%C A086543 (2111) (3111) (331)
%C A086543 (11111) (21111) (421)
%C A086543 (111111) (511)
%C A086543 (2221)
%C A086543 (3211)
%C A086543 (4111)
%C A086543 (22111)
%C A086543 (31111)
%C A086543 (211111)
%C A086543 (1111111)
%C A086543 (End)
%H A086543 Vincenzo Librandi, <a href="/A086543/b086543.txt">Table of n, a(n) for n = 0..300</a>
%F A086543 A000041(n) if n is odd; otherwise, A000041(n) - A000041(n/2).
%F A086543 G.f.: Sum_{k>=1} x^(2k-1)/((Product_{j=1..2k-1} (1-x^j))*(Product_{j>=k} (1-x^(2j)))). - _Emeric Deutsch_, Mar 30 2006
%F A086543 G.f.: 1/E(x) - 1/E(x^2) where E(x) = prod(n>=1, 1-x^n ); see Pari code. - _Joerg Arndt_, May 04 2013
%e A086543 a(4)=3 because we have [3,1],[2,1,1] and [1,1,1] ([4] and [2,2] do not qualify).
%p A086543 g:=sum(x^(2*k-1)/product(1-x^j,j=1..2*k-1)/product(1-x^(2*j),j=k..70),k=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..45); # _Emeric Deutsch_, Mar 30 2006
%t A086543 nn=50;CoefficientList[Series[Sum[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{k,1,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 28 2013 *)
%t A086543 Table[Length[Select[IntegerPartitions[n],Max[#]!=n/2&]],{n,0,30}] (* _Gus Wiseman_, Oct 12 2023 *)
%o A086543 (PARI) x='x+O('x^66); concat([0], Vec(1/eta(x)-1/eta(x^2)) ) \\ _Joerg Arndt_, May 04 2013
%Y A086543 Cf. A038348, A047967.
%Y A086543 The complement is counted by A035363, ranks A344415.
%Y A086543 These partitions have ranks A366322.
%Y A086543 A025065 counts partitions with sum <= twice length, ranks A344296.
%Y A086543 A110618 counts partitions with sum >= twice maximum, ranks A344291.
%Y A086543 Cf. A238628, A257991, A300061, A320924, A340387, A344414, A366319.
%K A086543 nonn
%O A086543 0,4
%A A086543 _Vladeta Jovovic_, Sep 10 2003