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%I #19 Oct 22 2023 16:10:46
%S 1,0,1,1,3,3,5,6,10,13,20,23,35,44,61,78,103,131,174,219,285,355,456,
%T 567,721,894,1117,1382,1718,2109,2607,3180,3902,4747,5789,7010,8500,
%U 10251,12373,14867,17868,21369,25584,30505,36372,43233,51350,60834,72039
%N Number of partitions of n containing a clique of size 4.
%C 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 Alois P. Heinz, <a href="/A183561/b183561.txt">Table of n, a(n) for n = 4..5000</a>
%F G.f.: (1-Product_{j>0} (1-x^(4*j)+x^(5*j))) / (Product_{j>0} (1-x^j)).
%e a(10) = 5, because 5 partitions of 10 contain (at least) one clique of size 4: [1,1,1,1,2,2,2], [1,1,2,2,2,2], [1,1,1,1,3,3], [1,1,1,1,2,4], [1,1,1,1,6].
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p add((l->`if`(j=4, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p end:
%p a:= n-> (l-> l[2])(b(n, n)):
%p seq(a(n), n=4..50);
%t max = 50; f = (1 - Product[1 - x^(4j) + x^(5j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x] , 4] (* _Jean-François Alcover_, Oct 01 2014 *)
%Y 4th column of A183568. Cf. A000041, A183558, A183559, A183560, A183562, A183563, A183564, A183565, A183566, A183567.
%K nonn
%O 4,5
%A _Alois P. Heinz_, Jan 05 2011
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