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A183562 Number of partitions of n containing a clique of size 5.

%I A183562 #19 Jun 12 2025 08:01:15
%S A183562 1,0,1,1,2,3,5,5,9,11,16,21,31,36,52,65,88,110,148,180,238,295,379,
%T A183562 469,600,731,926,1133,1413,1725,2141,2590,3194,3864,4719,5692,6924,
%U A183562 8301,10049,12026,14468,17263,20694,24586,29359,34804,41372
%N A183562 Number of partitions of n containing a clique of size 5.
%C A183562 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 A183562 Alois P. Heinz, <a href="/A183562/b183562.txt">Table of n, a(n) for n = 5..1000</a>
%F A183562 G.f.: (1-Product_{j>0} (1-x^(5*j)+x^(6*j))) / (Product_{j>0} (1-x^j)).
%F A183562 a(n) = A000041(n) - A184640(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183562 a(11) = 5, because 5 partitions of 11 contain (at least) one clique of size 5: [1,1,1,1,1,2,2,2], [1,2,2,2,2,2], [1,1,1,1,1,3,3], [1,1,1,1,1,2,4], [1,1,1,1,1,6].
%p A183562 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183562       add((l->`if`(j=5, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183562     end:
%p A183562 a:= n-> (l-> l[2])(b(n, n)):
%p A183562 seq(a(n), n=5..55);
%t A183562 max = 55; f = (1 - Product[1 - x^(5j) + x^(6j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 5] (* _Jean-François Alcover_, Oct 01 2014 *)
%t A183562 Table[Count[IntegerPartitions[n,{5,PartitionsP[n]}],_?(MemberQ[ Length/@ Split[ #],5]&)],{n,5,60}] (* _Harvey P. Dale_, Feb 02 2019 *)
%Y A183562 5th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183563, A183564, A183565, A183566, A183567.
%K A183562 nonn
%O A183562 5,5
%A A183562 _Alois P. Heinz_, Jan 05 2011