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 A183558 #38 Apr 26 2019 17:38:07
%S A183558 0,1,1,2,3,6,7,13,16,25,33,49,61,90,113,156,198,269,334,448,556,726,
%T A183558 902,1163,1428,1827,2237,2817,3443,4302,5219,6478,7833,9632,11616,
%U A183558 14197,17031,20712,24769,29925,35688,42920,50980,61059,72318,86206,101837,120941
%N A183558 Number of partitions of n containing a clique of size 1.
%C A183558 All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.
%H A183558 Alois P. Heinz, <a href="/A183558/b183558.txt">Table of n, a(n) for n = 0..5000</a>
%F A183558 G.f.: (1-Product_{j>0} (1-x^(j)+x^(2*j))) / (Product_{j>0} (1-x^j)).
%F A183558 From _Vaclav Kotesovec_, Nov 15 2016: (Start)
%F A183558 a(n) = A000041(n) - A007690(n).
%F A183558 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)
%e A183558 a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
%e A183558 From _Gus Wiseman_, Apr 19 2019: (Start)
%e A183558 The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
%e A183558 (1) (2) (3) (4) (5) (6) (7) (8)
%e A183558 (21) (31) (32) (42) (43) (53)
%e A183558 (211) (41) (51) (52) (62)
%e A183558 (221) (321) (61) (71)
%e A183558 (311) (411) (322) (332)
%e A183558 (2111) (3111) (331) (422)
%e A183558 (21111) (421) (431)
%e A183558 (511) (521)
%e A183558 (2221) (611)
%e A183558 (3211) (3221)
%e A183558 (4111) (4211)
%e A183558 (31111) (5111)
%e A183558 (211111) (32111)
%e A183558 (41111)
%e A183558 (311111)
%e A183558 (2111111)
%e A183558 (End)
%p A183558 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183558 add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183558 end:
%p A183558 a:= n-> b(n$2)[2]:
%p A183558 seq(a(n), n=0..50);
%t A183558 max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* _Jean-François Alcover_, Oct 01 2014. Edited by _Gus Wiseman_, Apr 19 2019 *)
%Y A183558 Column k=1 of A183568.
%Y A183558 Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
%Y A183558 Cf. A052485, A090858, A117571, A127002, A325241, A325242, A325244.
%K A183558 nonn
%O A183558 0,4
%A A183558 _Alois P. Heinz_, Jan 05 2011
%E A183558 a(0)=0 prepended by _Gus Wiseman_, Apr 19 2019