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A183559 Number of partitions of n containing a clique of size 2.

%I A183559 #24 Jun 12 2025 07:59:11
%S A183559 1,0,2,2,3,5,9,10,16,23,31,43,60,75,106,140,179,237,310,389,508,647,
%T A183559 815,1032,1305,1617,2033,2527,3117,3857,4764,5812,7142,8711,10585,
%U A183559 12866,15605,18803,22716,27325,32774,39286,47016,56019,66819,79456,94273,111766
%N A183559 Number of partitions of n containing a clique of size 2.
%C A183559 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 A183559 Alois P. Heinz, <a href="/A183559/b183559.txt">Table of n, a(n) for n = 2..1000</a>
%F A183559 G.f.: (1-Product_{j>0} (1-x^(2*j)+x^(3*j))) / (Product_{j>0} (1-x^j)).
%F A183559 a(n) = A000041(n) - A116645(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183559 a(7) = 5, because 5 partitions of 7 contain (at least) one clique of size 2: [1,1,1,2,2], [1,1,2,3], [2,2,3], [1,3,3], [1,1,5].
%p A183559 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183559       add((l->`if`(j=2, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183559     end:
%p A183559 a:= n-> (l-> l[2])(b(n, n)):
%p A183559 seq(a(n), n=2..50);
%t A183559 max = 50; f = (1 - Product[1 - x^(2j) + x^(3j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 2] (* _Jean-François Alcover_, Oct 01 2014 *)
%Y A183559 Column k=2 of A183568.
%Y A183559 Cf. A000041, A183558, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
%K A183559 nonn
%O A183559 2,3
%A A183559 _Alois P. Heinz_, Jan 05 2011