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A183560 Number of partitions of n containing a clique of size 3.

%I A183560 #19 Jun 12 2025 07:59:56
%S A183560 1,0,1,2,3,3,6,8,13,15,24,30,44,54,77,98,134,165,222,279,367,454,588,
%T A183560 731,936,1148,1454,1788,2241,2732,3400,4140,5106,6183,7579,9157,11156,
%U A183560 13406,16249,19482,23489,28042,33666,40087,47914,56851
%N A183560 Number of partitions of n containing a clique of size 3.
%C A183560 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 A183560 Alois P. Heinz, <a href="/A183560/b183560.txt">Table of n, a(n) for n = 3..1000</a>
%F A183560 G.f.: (1-Product_{j>0} (1-x^(3*j)+x^(4*j))) / (Product_{j>0} (1-x^j)).
%F A183560 a(n) = A000041(n) - A118807(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183560 a(9) = 6, because 6 partitions of 9 contain (at least) one clique of size 3: [1,1,1,2,2,2], [2,2,2,3], [1,1,1,3,3], [3,3,3], [1,1,1,2,4], [1,1,1,6].
%p A183560 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183560       add((l->`if`(j=3, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183560     end:
%p A183560 a:= n-> (l-> l[2])(b(n, n)):
%p A183560 seq(a(n), n=3..50);
%t A183560 max = 50; f = (1 - Product[1 - x^(3j) + x^(4j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 3] (* _Jean-François Alcover_, Oct 01 2014 *)
%Y A183560 Column k=3 of A183568.
%Y A183560 Cf. A000041, A183558, A183559, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
%K A183560 nonn
%O A183560 3,4
%A A183560 _Alois P. Heinz_, Jan 05 2011