cp's OEIS Frontend

A183561 Number of partitions of n containing a clique of size 4.

%I A183561 #21 Jun 12 2025 08:00:37
%S A183561 1,0,1,1,3,3,5,6,10,13,20,23,35,44,61,78,103,131,174,219,285,355,456,
%T A183561 567,721,894,1117,1382,1718,2109,2607,3180,3902,4747,5789,7010,8500,
%U A183561 10251,12373,14867,17868,21369,25584,30505,36372,43233,51350,60834,72039
%N A183561 Number of partitions of n containing a clique of size 4.
%C A183561 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 A183561 Alois P. Heinz, <a href="/A183561/b183561.txt">Table of n, a(n) for n = 4..5000</a>
%F A183561 G.f.: (1-Product_{j>0} (1-x^(4*j)+x^(5*j))) / (Product_{j>0} (1-x^j)).
%F A183561 a(n) = A000041(n) - A184639(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183561 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 A183561 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183561       add((l->`if`(j=4, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183561     end:
%p A183561 a:= n-> (l-> l[2])(b(n, n)):
%p A183561 seq(a(n), n=4..50);
%t A183561 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 A183561 4th column of A183568. Cf. A000041, A183558, A183559, A183560, A183562, A183563, A183564, A183565, A183566, A183567.
%K A183561 nonn
%O A183561 4,5
%A A183561 _Alois P. Heinz_, Jan 05 2011