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A183566 Number of partitions of n containing a clique of size 9.

%I A183566 #15 Jun 12 2025 08:03:51
%S A183566 1,0,1,1,2,2,4,4,7,9,13,15,23,27,38,47,63,77,103,126,165,201,258,315,
%T A183566 401,487,611,743,924,1118,1382,1664,2041,2455,2989,3583,4340,5185,
%U A183566 6248,7446,8930,10604,12668,15002,17848,21083,24987,29435,34776,40860
%N A183566 Number of partitions of n containing a clique of size 9.
%C A183566 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 A183566 Alois P. Heinz, <a href="/A183566/b183566.txt">Table of n, a(n) for n = 9..1000</a>
%F A183566 G.f.: (1-Product_{j>0} (1-x^(9*j)+x^(10*j))) / (Product_{j>0} (1-x^j)).
%F A183566 a(n) = A000041(n) - A184644(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183566 a(12) = 1, because 1 partition of 12 contains (at least) one clique of size 9: [1,1,1,1,1,1,1,1,1,3].
%p A183566 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183566       add((l->`if`(j=9, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183566     end:
%p A183566 a:= n-> (l-> l[2])(b(n, n)):
%p A183566 seq(a(n), n=9..60);
%t A183566 max=60;f=(1-Product[1-x^(9j)+x^(10j),{j,1,max}])/Product[1-x^j,{j,1,max}]; s=Series[f,{x,0,max}]; Drop[CoefficientList[s,x],9] (* _Jean-François Alcover_, Oct 01 2014 *)
%Y A183566 9th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183567.
%K A183566 nonn
%O A183566 9,5
%A A183566 _Alois P. Heinz_, Jan 05 2011