cp's OEIS Frontend

A183563 Number of partitions of n containing a clique of size 6.

%I A183563 #17 Jun 12 2025 08:01:59
%S A183563 1,0,1,1,2,2,5,5,8,10,15,18,27,33,47,57,78,96,129,159,208,258,330,407,
%T A183563 517,635,798,978,1217,1482,1833,2225,2729,3303,4028,4856,5885,7070,
%U A183563 8528,10211,12259,14628,17494,20800,24777,29378,34867
%N A183563 Number of partitions of n containing a clique of size 6.
%C A183563 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 A183563 Alois P. Heinz, <a href="/A183563/b183563.txt">Table of n, a(n) for n = 6..1000</a>
%F A183563 G.f.: (1-Product_{j>0} (1-x^(6*j)+x^(7*j))) / (Product_{j>0} (1-x^j)).
%F A183563 a(n) = A000041(n) - A184641(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183563 a(10) = 2, because 2 partitions of 10 contain (at least) one clique of size 6: [1,1,1,1,1,1,2,2], [1,1,1,1,1,1,4].
%p A183563 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183563       add((l->`if`(j=6, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183563     end:
%p A183563 a:= n-> (l-> l[2])(b(n, n)):
%p A183563 seq(a(n), n=6..55);
%t A183563 max = 55; f = (1 - Product[1 - x^(6j) + x^(7j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 6] (* _Jean-François Alcover_, Oct 01 2014 *)
%Y A183563 6th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183564, A183565, A183566, A183567.
%K A183563 nonn
%O A183563 6,5
%A A183563 _Alois P. Heinz_, Jan 05 2011