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%I #17 Aug 12 2018 17:12:46
%S 1,0,1,1,2,2,4,4,8,9,13,16,24,28,40,49,66,82,110,132,175,214,274,336,
%T 428,520,655,798,990,1203,1486,1793,2200,2653,3227,3880,4701,5622,
%U 6779,8092,9701,11546,13793,16355,19466,23029,27290,32199,38048,44752,52719
%N Number of partitions of n containing a clique of size 8.
%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="/A183565/b183565.txt">Table of n, a(n) for n = 8..1000</a>
%F G.f.: (1-Product_{j>0} (1-x^(8*j)+x^(9*j))) / (Product_{j>0} (1-x^j)).
%e a(12) = 2, because 2 partitions of 12 contain (at least) one clique of size 8: [1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,4].
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p add((l->`if`(j=8, [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=8..60);
%t max = 60; f = (1 - Product[1 - x^(8j) + x^(9j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 8] (* _Jean-François Alcover_, Oct 01 2014 *)
%t c8[n_]:=If[MemberQ[Tally[n][[All,2]],8],1,0]; Table[Total[c8/@ IntegerPartitions[ x]],{x,8,60}] (* _Harvey P. Dale_, Aug 12 2018 *)
%Y 8th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183566, A183567.
%K nonn
%O 8,5
%A _Alois P. Heinz_, Jan 05 2011
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